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TRANSCRIPT
Real and Complex
Dynamical Systems
International conference on the occasion ofProf. Yulij Ilyashenko’s 75th birthday
Moscow, November 26–30, 2018
Organizers
Founding and sponsoring institutions
The Interdisciplinary Scientific Center J.-V. PonceletNational Research University Higher School of EconomicsSteklov Mathematical Institute of RASIndependent University of Moscow
Russian Foundation for Basic Research
Program Committee
Alexander Bufetov (chair)Alexey GlutsyukFrank LorayYakov SinaiSergey Voronin
Organizing Committee
Dmitry FilimonovAlexey Glutsyuk (chair)Anton GorodetskiAlexey KlimenkoIlya SchurovAlexandra SkripchenkoVladlen TimorinMichael Tsfasman
http://rcds.dyn-sys.org/
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Contents
Talks 1
Beloshapka, Iulia. Irreducible representations offinitely generated nilpotent groups . . . . . . . 1
Binyamini, Gal. Geometry of Foliations over Num-ber Fields and Applications . . . . . . . . . . . 2
Bjerklov, Kristian. Positive Lyapunov exponents forquasi-periodically driven circle endomorphismswith critical points . . . . . . . . . . . . . . . . 3
Boulanger, Adrien. Heat kernel and counting prob-lems for Kleinian groups. . . . . . . . . . . . . 3
Buchstaber, Victor and Glutsyuk, Alexey. On phase-lock areas in model of Josephson effect and dou-ble confluent Heun equations . . . . . . . . . . 4
Dehornoy, Pierre. Classification of surfaces trans-verse to flows: the case of the geodesic flow . . 6
Deroin, Bertrand. Structural stability in the mod-uli space of algebraic foliations of the complexprojective plane . . . . . . . . . . . . . . . . . . 6
Gorodetski, Anton. Sums of Cantor sets in spectraltheory . . . . . . . . . . . . . . . . . . . . . . . 7
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Grines, Vyacheslav. On hyperbolic attractors andtopology of ambient manifolds . . . . . . . . . . 8
Guillot, Adolfo. On the multipliers at fixed points ofrational self-maps of the projective plane . . . . 9
Guirardel, Vincent. Surface groups of germs of ana-lytic diffeomorphisms in one variable . . . . . . 10
Hubert, Pascal. Normal distribution of correlationmeasures of binary sum-of-digits functions . . . 10
Isaenkova, Natalia. A generalization of the diffeo-morphism by Smale with hyperbolic attractor . 11
Khanin, Konstantin. Schrodinger operators withquasi-periodic potentials and the Aubry-Mather theory . . . . . . . . . . . . . . . . . . 12
Kleptsyn, Victor. Furstenberg theorem: now with aparameter! . . . . . . . . . . . . . . . . . . . . . 13
Klimenko, Alexey. Convergence of spherical averagesfor actions of Fuchsian Groups . . . . . . . . . 14
Kuksin, Sergei. Ergodicity, mixing and KAM . . . . 16
Lerman, Lev. On skew products over a quasi-periodicshift of the circle . . . . . . . . . . . . . . . . . 17
Matheus, Carlos. Lagrange and Markov are very dif-ferent from Hausdorff’s point of view . . . . . . 18
Medvedeva, Nataliia. Construction of formulas forthe coefficients of the asymptotic expansion ofthe correspondence map and the monodromytransformation . . . . . . . . . . . . . . . . . . 18
Mironov, Andrey. Integrable magnetic geodesic flowson 2-torus: new example via quasi-linear systemof PDEs . . . . . . . . . . . . . . . . . . . . . . 19
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Novikov, Dmitry and Binyamini, Gal. Complex Cel-lular Structures . . . . . . . . . . . . . . . . . . 20
Okunev, Alexey. A 𝐶1 Anosov diffeomorphism witha horseshoe that attracts almost any point . . . 21
Ortiz-Bobadilla, Laura and Juarez, Jessica-Rosas.Analytic invariants of germs of curves and fo-liations in (C2, 0) . . . . . . . . . . . . . . . . . 22
Paris-Romaskevich, Olga. Negative refraction tilingbilliards . . . . . . . . . . . . . . . . . . . . . . 23
Pochinka, Olga. On Palis Problem of Embedding ofMorse-Smale Cascades into Flows . . . . . . . . 24
Polotovskiy, Grigory. Dmitry Andreevich Gud-kov – an outstanding mathematician andthe researcher of Lobachevsky’s biography(to the 100 anniversary since birth) . . . . . . . 26
Qiu, Yanqi. Patterson–Sullivan construction forpoint processes and the reconstruction for har-monic functions . . . . . . . . . . . . . . . . . . 27
Rebelo, Julio. Global approximation on Diff𝜔(𝑆1) . . 28
Reis, Helena. On reduction of singularities for vec-tor fields in dimension 3 . . . . . . . . . . . . . 29
Rosales-Gonzalez, Ernesto. On the classification ofgerms of foliations in (𝐶𝑛, 0) . . . . . . . . . . 30
Saprykina, Maria. Erratic behavior for one-dimensional random walks in a generic quasi-periodic environment . . . . . . . . . . . . . . . 32
Shilin, Ivan. Sparkling saddle loops of vector fieldson surfaces . . . . . . . . . . . . . . . . . . . . 33
Skripchenko, Sasha. Cohomological equations for lin-ear involutions . . . . . . . . . . . . . . . . . . 34
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Solodovnikov, Nikita. Bifurcations of families of vec-tor fields on the two-dimensional sphere . . . . 35
Stolovitch, Laurent. Holomorphic normal form ofnonlinear perturbations of nilpotent vector fields 36
Treschev, Dmitry. On a heavy quantum particle . . 37
Zhuzhoma, Evgeny. On classification of Morse-Smaledynamical systems with few non-wandering points 38
Zorich, Anton. Flat and hyperbolic enumerative ge-ometry . . . . . . . . . . . . . . . . . . . . . . . 38
Posters 41
Barinova, Marina. Energy function for 3-diffeomorphisms with one-dimensional surfaceattractor and repeller . . . . . . . . . . . . . . 41
Berezin, Sergey and Bufetov, Alexander. CentralLimit Theorem for Laguerre Ensembles UnderDouble Scaling . . . . . . . . . . . . . . . . . . 43
Bogaevsky, Ilya. Difference between smooth and for-mal classifications of Morse transitions of im-plicit ODE . . . . . . . . . . . . . . . . . . . . 43
Fierobe, Corentin. On the circumcenters of triangu-lar orbits in elliptic billiard and complex reflex-ion law . . . . . . . . . . . . . . . . . . . . . . . 45
Fomichov, Vladimir. Evolution of moments ofisotropic Brownian stochastic flows . . . . . . . 46
Kruglov, Vladislav. Topological conjugacy of Ω-stable flows without limit cycles on surfaces . . 47
Kurenkov, Evgeny. Classification of one-dimensionalattractors by means of Pseudo-Anosov homeo-morphisms . . . . . . . . . . . . . . . . . . . . . 49
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Mamayusupov, Khudoyor. The space of cubic New-ton maps with persistent parabolics . . . . . . 51
Marshall, Juan. Modulus of continuity away fromzero for correlation measures in substitutiontiling systems . . . . . . . . . . . . . . . . . . . 52
Molina-Samper, Beatriz. Invariant surfaces for Torictype foliations in dimension 3 . . . . . . . . . . 54
Monakov, Grigorii. On displacement of viscous liq-uid in a quasi 1D system . . . . . . . . . . . . . 55
Nikitin, Pavel. On Number Rigidity for Pfaffianpoint processes . . . . . . . . . . . . . . . . . . 56
Nozdrinova, Elena. On periodic data of diffeomor-phisms with one saddle orbit . . . . . . . . . . 56
Paziy, Natalia. Normal forms of infinite indextransversal systems with one consrtraint . . . . 59
Perstneva, Polina. Periodic solutions of one-dimensional parabolic problems with hysteresison the boundary . . . . . . . . . . . . . . . . . 60
Pilyugin, Sergei. Yu. S. Ilyashenko – from local toglobal . . . . . . . . . . . . . . . . . . . . . . . 61
Rakhimov, Karim. Dynamics of geodesics of mero-morphic connections on compact Riemann sur-faces . . . . . . . . . . . . . . . . . . . . . . . . 62
Shaikhullina, Polina. Strict analytic classification ofsimplest germs of semi-hyperbolic maps . . . . 63
Smilga, Ilia. Milnorian and non-Milnorian represen-tations . . . . . . . . . . . . . . . . . . . . . . . 65
Tikhomirov, Sergey. Various problems in parabolicequations with hysteresis . . . . . . . . . . . . . 65
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Zo ladek, Henryk. The Weotzenbock theorem aboutinvariants . . . . . . . . . . . . . . . . . . . . . 66
Zubov, Dmitry. Finitely additive measures on theinvariant foliations of Anosov diffeomorphisms . 67
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Talks
Irreducible representations offinitely generated nilpotent groups
Iulia Beloshapka
ETH, Zurich, Switzerland
At ICM 2010 Parshin conjectured that irreducible complexrepresentations of finitely generated nilpotent groups aremonomial if and only if they have finite weight. Every ir-reducible complex representation is known to be monomial forfinite nilpotent groups and for unitary irreducible representa-tions of connected nilpotent Lie groups (due to A.A. Kirillovand J. Dixmier). We prove Parshin’s conjecture in full gener-ality. We will also discuss how to approach the moduli spaceproblem for such representations, and its relation to higher-dimensional adeles. For nilpotency classes 2 and 3 we obtaineda full description of the coarse moduli space.
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Geometry of Foliations over Number Fields andApplications
Gal Binyamini
Weizmann Institute of Science
Many problems related to the study of limit cycles and theHilbert 16th problem are equivalent to counting the roots forcertain types of (univariate) analytic functions. One approachto such zero-counting problems is to compare the number ofzeros of an analytic function with its asymptotic growth usingthe Jensen formula. Yulij Ilyashenko has been a champion ofthis approach, applying it together with his students to severalclasses of functions including solutions of linear analytic ODEs,abelian integrals and Poincare maps of planar systems.
In this talk I will discuss recent progress on higher dimen-sional analogs of the growth-vs-zeros method, i.e. studying so-lutions of systems of equations in several variables by lookingat their asymptotic growth. More specifically, I will show howthis technique leads to good bounds when the equations con-sist of functions satisfying overdetermined systems of algebraicPDEs over number fields. This class notably contains manyfunctions of importance in arithmetic geometry including el-liptic, abelian, modular and Riemann theta functions. I willalso sketch without much technical detail how applying thistechnique to such functions one can obtain effective estimatesfor various problems in diophantine geometry (e.g. Manin-Mumford, Andre-Oort, Masser-Zannier and Katz-Oort).
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Positive Lyapunov exponents forquasi-periodically driven circle endomorphisms
with critical points
Kristian Bjerklov
KTH, Stockholm
We consider skew-products over irrational rotation on the cir-cle, where the fiber maps are circle endomorphisms which havea unique critical point. For a class of such maps (whose criti-cal points are “non-flat”) we show that the (fibered) Lyapunovexponents are positive for a.e. initial condition.
This is a joint work with H. Eliasson.
Heat kernel and counting problems for Kleiniangroups.
Adrien Boulanger
Universite Aix-Marseille
Given a group acting discretely and properly on a metric space,we are interested in estimating the number of points belongingto both the orbit of a given point and to a ball of fixed centerwhen the radius of the ball goes to infinity. Any control of thisasymptotic is a counting theorem.
In this talk we will focus on Kleinian groups, i.e groupsacting on a hyperbolic space. We will start with the historicalbackground, and proceed with the proposal of a strategy toobtain counting theorems based on the use of the heat kernel.
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On phase-lock areas in model of Josephson effectand double confluent Heun equations
Victor Buchstaber 𝑎 and Alexey Glutsyuk 𝑏
𝑎 Steklov Mathematical Institute (Moscow) and VNIIFTRI(Mendeleevo),
𝑏 National Research University Higher School of Economics (HSE),Moscow, Russia,
𝑏 CNRS, France (ENS de Lyon and ISC Poncelet)𝑎 [email protected], 𝑏 [email protected]
In 1973 B. Josephson received Nobel Prize for discovering anew fundamental effect in superconductivity concerning a sys-tem of two superconductors separated by a very narrow di-electric: there could exist a supercurrent tunneling throughthis junction (called Josephson junction). One of the knownmodels of the overdamped Josephson junction used a familyof non-linear first order differential equations. Motivated byproblems on Josephson effect, V. M. Buchstaber, O. V. Karpovand S. I. Tertychnyi studied this family in a series of papers(see JETP 2001 and later). They introduced and studied anequivalent family of dynamical systems on 2-torus. Motivatedby dynamical systems point of view, Yu. S. Ilyashenko andJ. Guckenheimer suggested and studied a subfamily of thisfamily on 2-torus as an interesting slow-fast system in 2001.Analogous systems arose in different models of mathematics,mechanics and physics that a priori were not related one tothe other.
The classical rotation number of the family of dynamicalsystems on 2-torus is a function of parameters of the family(identified with the average voltage through a long time in-terval). The 𝑟-th level set of the rotation number function
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is called the 𝑟-th phase-lock area, if it has non-empty inte-rior. Our family is very atypical: the phase-lock areas existonly for integer values of the rotation number, in differenceto the classical Arnold tongue picture (the quantization effectdiscovered by V. M. Buchstaber, O. V. Karpov and S. I. Ter-tychnyi in 2010). The complexification of our family is theprojectivization of a family of classical double confluent Heunequations: second order linear differential equations on theRiemann sphere with two irregular singular points at 0 andinfinity.
In the talk we will present results on geometric and analyticproperties of the boundaries of the phase-lock areas. We willdiscuss several important geometric conjectures on the phase-lock areas and analytic conjectures on Heun equations and re-lations between them. We will present the state of art in thisarea including recent works of V. M. Buchstaber, S. I. Ter-tychnyi, A. A. Glutsyuk, V. A. Kleptsyn, D. A. Filimonov,I. V. Schurov.
Acknowledgements. Research of V. M. Buchstaber wassupported by Russian Foundation for Basic Research (RFBR)grants 14-01-00506, 17-01-00192. Research of A. A. Glutsyukwas supported by Russian Science Foundation (RSF) grant 18-41-05003, by Russian Foundation for Basic Research (RFBR)grants 13-01-00969-a, 16-01-00748, 16-01-00766, and by FrenchNational Research Agency (ANR) grant ANR-13-JS01-0010.
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Classification of surfaces transverse to flows:the case of the geodesic flow
Pierre Dehornoy
Universite de Grenoble–Alpes 1, France
Given a non singular vector field, transverse hypersurfaces(when they exist) are good tools to study the flow dynam-ics. The goal of this talk is, in dimension 3 and for the par-ticular case of the geodesic flow on a hyperbolic surface, toclassify such surfaces. For this, we introduce an elementarytool, the intersection norms, associated to a finite collection ofclosed curves on a surface. These norms are simple cousins ofthe Thurston-Gromov norms and, like their cousins, their dualunit balls are integer polytopes. The integer points in thesepolytopes yield the classification we are looking for.
Structural stability in the moduli space ofalgebraic foliations of the complex projective
plane
Bertrand Deroin
Universite de Cergy-Pontoise
I will prove the existence of a non trivial stability componentin the space of degree two algebraic foliations of the complexprojective plane, consisting of foliations having a non trivialFatou component: a fibration by discs over a Riemann surfaceof genus three. This result can be opposed to the famous re-sult of Yu. Ilyashenko (adressed at the ICM 78) establishing
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minimality, ergodicity and rigidity for generic foliations of theplane that preserve a line, and its generalizations by F. Lo-ray/J. Rebelo or T. Golenishcheva-Kutusova/V. Kleptsyn.
Sums of Cantor sets in spectral theory
Anton Gorodetski
University of California Irvine
Structure of sums of Cantor sets is a classical question relatedto many problems in dynamical systems, number theory, andgeometric measure theory. We will provide an overview of cur-rently known results and open questions, and discuss numerousrecent applications to spectral theory of Schrodinger operators(the results of joint projects with D. Damanik, J. Fillman, andB. Solomyak).
REFERENCES
[1] D. Damanik, A. Gorodetski, Spectral Transitions for the SquareFibonacci Hamiltonian, Journal of Spectral Theory, vol. 8(2018), pp. 1487–1507.
[2] D. Damanik, J. Fillman, A. Gorodetski, MultidimensionalAlmost-Periodic Schrodinger Operators with Cantor Spec-trum, preprint (arXiv:1809.02720).
[3] D. Damanik, A. Gorodetski, B. Solomyak, Absolutely Contin-uous Convolutions of Singular Measures and an Applicationto the Square Fibonacci Hamiltonian, Duke Math. J., vol. 164(2015), pp. 1603–1640.
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On hyperbolic attractors andtopology of ambient manifolds
Vyacheslav Grines
National Research University. Higher School of Economics
According to S. Smale, Ya. Sinai, J. Franks, S. Newhouse andE. Manning, a manifold 𝑀𝑛 which admits codimension oneAnosov diffeomorphism is the torus. A similar result is ob-tained By V. Z. Grines and E. V. Zhuzhoma in the case whenthe non-wandering set of structurally stable diffeomorphismcontains orientable expanding attractor of codimension one(see [1] for information and references). In [2] the topologi-cal classification of three-dimensional manifolds have been ob-tained for manifolds admitting structurally stable diffeomor-phisms whose nonwandering set consists of two-dimensionalattractors and repellers. Moreover it has been obtained com-plete topological classification of described above classes ofdiffeomorphisms.
The report is supported by the HSE Basic Research Pro-gram (project no. 95) in 2018.
REFERENCES
[1] V. Z. Grines, Ye. V. Zhuzhoma, and O. V. Pochinka. Roughdiffeomorphisms with basic sets of codimension one. Journal ofMathematical Sciences, Vol. 225, No. 2, August, 2017. 195-218. V.S.
[2] V. Grines, O. Pochinka, Yu. Levchenko, V. S. Medvedev Thetopological classification of structural stable 3-diffeomorphismswith two-dimensional basic sets // Nonlinearity. 2015. Vol. 28.P. 4081-4102.
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On the multipliers at fixed points of rationalself-maps of the projective plane
Adolfo Guillot
Instituto de Matematicas, UNAM, Mexico
For a rational self-map of the complex projective space of di-mension 𝑛, 𝑓 : P𝑛 → P𝑛, we may consider the collection of theeigenvalues of its derivative at its fixed points. It depends onlyon the linear equivalence class of 𝑓 . These numbers cannot bearbitrary. For instance, the holomorphic Lefschetz fixed-pointtheorem gives an algebraic relation between them, and moreof these can be obtained through the Atiyah-Bott fixed-pointtheorem. However, a parameter-counting argument shows thatthere are algebraic relations among these multipliers that arenot obtained through these theorems. What do we know ofthese relations? Can we give them explicitly? To what extentdoes the collection of the eigenvalues at the fixed points deter-mine the linear equivalence class of a rational self-map? Wedo not have answers to these questions even in the simplestcase, the one where 𝑛 = 2 and where 𝑓 is given by quadraticpolynomials.
We will discuss these problems for those rational maps ofthe projective plane which are given by quadratic polynomialsand which have, moreover, a forward-invariant line. It is ajoint work with Valente Ramırez (Rennes).
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Surface groups of germs of analyticdiffeomorphisms in one variable
Vincent Guirardel
Universite de Rennes 1
Let Diff(C, 0) be the group of germs of analytic diffeomor-phisms at the origin 0 ∈ C, i.e. of power series in one variablewith positive radius of convergence and with non-zero deriva-tive at 0. We prove that this group contains subgroups isomor-phic to the fundamental group of any closed orientable surface.This is a joint work with Cantat, Cerveau and Souto.
Normal distribution of correlation measures ofbinary sum-of-digits functions
Pascal Hubert
Aix-Marseille University
I will explain a joint result with Jordan Emme. We studycorrelation measures for the sum of digits in base 2. Denote by𝜇𝑎(𝑑) the asymptotic density of the set ℰ𝑎,𝑑 = {𝑛 ∈ N, 𝑠2(𝑛+𝑎) − 𝑠2(𝑛) = 𝑑} (where 𝑠2 is the sum-of-digits function inbase 2). Then, for any point 𝑋 in {0, 1}N, define the integersequence (𝑎𝑋(𝑛))𝑛∈N such that the binary decomposition of𝑎𝑋(𝑛) is the prefix of length 𝑛 of 𝑋. We prove that for anyshift-invariant ergodic probability measure 𝜈 on {0, 1}N, thesequence
(𝜇𝑎𝑋(𝑛)
)𝑛∈N satisfies a central limit theorem.
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A generalization of the diffeomorphism by Smalewith hyperbolic attractor
Natalia Isaenkova
National Research University Higher School of Economics
Smale in 1967 [1] proposed a method of constructing diffeo-morphisms with hyperbolic attractors using a expanding en-domorphism. This construct was implemented explicitly usingexpanding endomorphism of the circle so that the result is athree-dimensional diffeomorphisms of the torus with solenoidalone-dimensional expanding attractor which is locally homeo-morphic to the product of the cut and Cantor set. We offer ageneralization of this construction, where instead of expandingendomorphism of the circle is regarded As the endomorphismof the (multidimensional) torus [2, 3]. Examines the type oftransitive invariant sets of the corresponding diffeomorphisms.
REFERENCES
[1] Smale S. Differentiable dynamical systems. Bull. Amer. Math. Soc.,1967, vol. 73, pp. 747–817.
[2] Zhuzhoma E. V., Isaenkova N. V. Zero-dimensional solenoidal basesets. Sb. Math., 2011, vol. 202, no. 3, pp. 351—372.
[3] Zhuzhoma E. V., Isaenkova N. V. On spectral decompositionof Smale-Vietoris axiom A diffeomorphisms. Dynamical Systems,2017, vol. 32, no. 2, pp. 221—233.
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Schrodinger operators with quasi-periodicpotentials and the Aubry-Mather theory
Konstantin Khanin
University of Toronto
Schrodinger operators with quasi-periodic potentials were in-tensively studied in the last few decades. Their spectral prop-erties depend on the value of the coefficient in front of thepotential, so called. coupling constant. For small values ofthe coupling constant the spectrum is absolutely continuous,while for large coupling constants the spectrum is pure point.Natural families of Schrodinger operators with quasi-periodicpotentials appear in the context of the Aubry-Mather the-ory. In this setting there are no coipling constants. Insteadoperators depend on the nonlinearity parameter for relatedarea-preserving maps. We shall discuss the transition fromthe absolutely continuous to the pure point spectrum for suchfamilies of Schrodinger operators.
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Furstenberg theorem: now with a parameter!
Victor Kleptsyn
CNRS, University of Rennes 1.
Consider a random product of i.i.d. matrices, randomly chosenfrom SL(2,R):
𝑇𝑛 = 𝐴𝑛...𝐴2𝐴1,
where the random matrices 𝐴𝑖 are i.i.d..A classical Furstenberg theorem then implies, that under
some very mild nondegeneracy conditions (no finite commoninvariant set of lines, no common invariant metric) for thelaw of 𝐴𝑖’s the norm of such a product almost surely growsexponentially.
Now, what happens if each of these matrices 𝐴𝑖(𝑠) dependson an additional parameter 𝑠, and hence so does their product𝑇𝑛(𝑠)? For each individual 𝑠, the Furstenberg theorem is stillapplicable. However, what can be said almost surely for therandom products 𝑇𝑛(𝑠), depending on a parameter?
We will impose a few reasonable additional assumptions,of which the most important is that the dependence of angleis monotonous w.r.t. the parameter: increasing the parameter“rotates all the directions clockwise”.
It turns out that, under these assumptions,
∙ Almost surely for all the parameter values, except for azero Hausdorff dimension (random) set, the Lyapunovexponent exists and equals to the Furstenberg one.
∙ Almost surely for all the parameter values the upper Lya-punov exponent equals to the Furstenberg one
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∙ At the same time, in the no-uniform-hyperbolicity pa-rameter region there exists a dense subset of parameters,for each of which the lower Lyapunov exponent takes anyfixed value between 0 and the Furstenberg exponent.
Our results are related to the Anderson localization in di-mension one, providing a purely dynamical viewpoint on itsproof.
This is a joint work [1] with Anton Gorodetski.
REFERENCES
[1] A. Gorodetski, V. Kleptsyn, Parametric Furstenberg The-orem on Random Products of SL(2,R) matrices, preprintarXiv:1809.00416.
Convergence of spherical averages for actions ofFuchsian Groups
Alexey Klimenko
Steklov Mathematical Institute of RAS,NRU Higher School of Economics
The talk is based on the joint work [1] with A. Bufetov andC. Series.
Consider an action of a Fuchsian group 𝐺 on a Lebesgueprobability space (𝑋,𝜇) by measure-preserving maps 𝑇𝑔. Letℛ be a fundamental domain of 𝐺 and let 𝐺0 be defined asabove. Assume that ℛ has even angles, that is, the bound-ary of the tesselation {𝑔ℛ} is a union of complete geodesic
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lines. Finally, for 𝜙 ∈ 𝐿1(𝑋,𝜇) define its spherical averages asfollows:
S𝑛(𝜙) =1
#{𝑔 : |𝑔| = 𝑛}∑
𝑔:|𝑔|=𝑛
𝜙 ∘ 𝑇𝑔,
where |𝑔| is a norm of 𝑔 ∈ 𝐺 with respect to the generatingset 𝐺0.
Theorem 1. Let 𝐺 be a non-elementary Fuchsian group 𝐺and suppose it has a fundamental domain ℛ with even cornersand satisfying a technical condition (in particular, it holds forany domain with at least 5 sides). Let 𝐺0 be a symmetric setof transformations defined above.
Let 𝐺 act on a Lebesgue probability space (𝑋,𝜇) bymeasure-preserving transformations. Denote by ℐ𝐺2
0the 𝜎-
algebra of sets invariant under all maps 𝑇𝑔1𝑔2, 𝑔1, 𝑔2 ∈ 𝐺0.Then, for any function 𝑓 ∈ 𝐿 log𝐿(𝑋,𝜇), the sequence(S2𝑛(𝑓)) converges as 𝑛 → ∞ almost surely and in 𝐿1 to theconditional expectation E(𝑓 |ℐ𝐺2
0) with respect to the 𝜎-algebra
ℐ𝐺20.
REFERENCES
[1] A. I. Bufetov, A. Klimenko and C. Series, Convergence ofspherical averages for actions of Fuchsian Groups, preprint,arXiv:1805.11743.
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Ergodicity, mixing and KAM
Sergei Kuksin
Universite Paris – 7
To prove the ergodicity for Hamiltonian systems of big orinfinite dimension is a notoriously complicated problem ofhigh importance. But what we often have in physics are not
Hamiltonian systems, but systems of the form <Hamiltonian
system> + <small dissipation> + <small random forcing>,where the random forcing may be very degenerate (i.e. itaffects only a few modes). For such systems an analogy ofthe ergodicity is called the mixing. In my talk I will remindthe definition of the mixing and explain how the KAM-theoryprovides a powerful tool to prove the mixing for the systemsabove. The talk is based on a joint work with Armen Shirikyanand Vahagn Nersesyan [1]
REFERENCES
[1] Nersesyan, V.; Shirikyan, A.R. Kuksin, S.B. Exponential mixingfor a class of dissipative PDEs with bounded degenerate noise.Preprint https://arxiv.org/abs/1802.03250
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On skew products over a quasi-periodicshift of the circle
Lev Lerman
Lobachevsky State University of Nizhny Novgorod
As is known, skew products over irrational shift on the circlearise as Poincare mappings for quasi-periodic nonautonomoussystems with two incommensurable frequencies. Many resultsfor such skew products exists when the related bundle overcircle is linear (reducibility, localization, spectral properties,etc.). Much work was done for the structure of 1-dimensionalskew products (with 1-dimensional leaves): non-smooth bifur-cations, rotation sets, and so forth. In this talk I would like totouch the topic of structural dynamics for such skew products,including classification problems.
The first topic is the dynamics of skew products with thecircle as a leaf. Here there is a simple way to distinguish ver-tically hyperbolic diffeomorpisms. For such diffeomorphismsthe classifying theorem is valid: the rotation number for ashift, the integer and some permutation is the complete set ofinvariants. Examples are presented when absence of verticalhyperbolicity leads to a dimension inhomogeneity of the in-variant sets. Analogs of such skew products in greater dimen-sions will be discussed (analogs of gradient-like skew products,Morse-Smale products).
The situation is much more involved when the leaves aren-dimensional submanifolds (𝑛 ≥ 2) and intersection of non-wandering set with leaves are infinite. Then even in the case ofvertical hyperbolicity the dynamics can be very complicated.Related examples will be discussed.
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Lagrange and Markov are very different fromHausdorff’s point of view
Carlos Matheus
CNRS and Ecole Polytechnique
The Lagrange and Markov spectra are subsets of R arisingnaturally in the study of certain Diophantine approximationproblems, heights of cusp excursions of geodesics of the mod-ular surface, etc.
After reviewing several features of these spectra, we willdiscuss our joint work with C. G. Moreira on the differencebetween them: in particular, we will describe our solution toa conjecture of Cusick from 1975.
Construction of formulas for the coefficients ofthe asymptotic expansion of the correspondence
map and the monodromy transformation
Nataliia Medvedeva
Chelyabinsk State University
An algorithm for distinguishing between stable and unstablefocus has long been known by calculating the coefficients ofthe monodromy transform series in the case when the singularpoint has a Newton diagram consisting of one edge. In thecase when the Newton diagram of a vector field consists ofseveral edges, the monodromy transformation can be decom-posed into a composition of so-called extended correspondence
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mappings, each of which corresponds to the inner vertex of theNewton diagram. An algorithm for constructing formulas forthe coefficients of the asymptotic decomposition (of arbitrarylength) of the extended correspondence mapping is developedin the absence of some resonances of the saddle obtained as aresult of a blow-up with a center at this internal vertex. Thepower of a set of exceptional resonances depends on the lengthof the asymptotic expansion. The constructed algorithm canbe implemented on a computer.
Partially supported by RFBR (project 17-01-00739-a)
Integrable magnetic geodesic flows on 2-torus:new example via quasi-linear system of PDEs
Andrey Mironov
Sobolev Institute of Mathematics, Novosibirsk
The only one example has been known of magnetic geodesicflow on the 2-torus which has a polynomial in momenta in-tegral independent of the Hamiltonian. In this example theintegral is linear in momenta. In the talk the problem of inte-grability on one energy level will be considered. This problemcan be reduced to a remarkable Semi-hamiltonian system ofquasi-linear PDEs and to the question of existence of smoothperiodic solutions for this system. Our main result states thatthe pair of Liouville metric with zero magnetic field on the2-torus can be analytically deformed to a Riemannian metricwith small magnetic field so that the magnetic geodesic flowon an energy level is integrable by means of a quadratic in mo-menta integral. The result were obtained with Michael Bialy(Tel Aviv) and Sergey Agapov (Novosibirsk).
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Complex Cellular Structures
Gal Binyamini and Dmitry Novikov
Weizmann Institute of Science
Real semialgebraic sets admit so-called cellular decomposi-tion, i.e. representation as a union of cells diffeomorphic tocubes, with some very convenient properties. Attempts tobuild a straightforward complex holomorphic generalizationof this construction meet difficulties related to inner metricproperties of holomorphic sets, absent in real case.
I will explain these difficulties and propose a proper com-plex generalization. The construction is motivated by and re-lates to uniformization results of resolution of singularities the-ory, by “growth-zeros” ideology used in Infinitesimal Hilbert16th problem and by Yomdin—Gromov parameterization re-sults. As a corollary, we solve an old Yomdin’s conjectureabout tail entropy and prove new results in transcendentalnumber theory.
https://arxiv.org/abs/1802.07577
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A 𝐶1 Anosov diffeomorphism with a horseshoethat attracts almost any point
Alexey Okunev
Yandex
Joint work with C. Bonatti, S. Minkov, and I. Shilin.
Using Bowen’s construction of a horseshoe with positiveLebesgue measure ([1]), we construct an example of a 𝐶1
Anosov diffeomorphism of the 2-torus that admits a physicalmeasure 𝜈 such that
∙ the basin of 𝜈 has full Lebesgue measure,
∙ 𝜈 is supported on a horseshoe ℋ with zero Lebesguemeasure.
The horseshoe ℋ is semithick, i.e. it is a product of a Can-tor set with positive measure in the unstable direction and aCantor set with zero measure in the stable direction.
REFERENCES
[1] Bowen R. A horseshoe with positive measure Inventiones mathemat-icae. 1975. 29, 3, 203-204.
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Analytic invariants of germs of curves andfoliations in (C2, 0)
Laura Ortiz-Bobadilla, Jessica Jaurez-Rosas
Instituto de Matematicas,Universidad Nacional Autonoma de Mexico (UNAM)
This talk deals with two geometric objects: on one side, thegerms of analytic curves in (C2, 0) and their analytic moduli,and on the other, the analytic functional invariants of germsof holomorphic dicritic foliations in (C2, 0). We prove thatany analytic class of germs of singular curves in (C2, 0) hav-ing 𝑛+1 pairwise transversal smooth branches can be realizedin an appropriate dicritic foliation with prescribed collectionof involutions. This collection of involutions is related to thetangency points with the exceptional divisor of the correspond-ing dicritic foliation; we recall that the involutions (with fixedparametrization) of a generic dicritic foliation is one of the,so called, Thom’s invariants (the functional one) of analyticclassification of generic dicritic foliations.
The purpose of this talk is to talk about a recent work withJessica Jaurez [1], and to give as well an overview of resultspreviously obtained in joint works with Ernesto Rosales andSergei Voronin [2, 3]. All together they constitute some stepstowards the thorough understanding of the geometry of thefinite parametric (analytic) invariants of singular curves andsingular foliations (work in progress).
REFERENCES
[1] J. Jaurez-Rosas, L. Ortiz-Bobadilla, Invariants of curves and pre-scribed involutions in germs of degenerated dicritic foliations in
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(C2, 0), preprint.
[2] L. Ortiz-Bobadilla, E. Rosales-Gonzalez, S. M. Voronin,Rigidity Theorems for Generic Holomorphic Germs of DicriticFoliations and Vector Fields in (C2, 0), Moscow MathematicalJournal, Vol. 5 , Number 1; 2005; pp. 171-206.
[3] L. Ortiz-Bobadilla, E. Rosales-Gonzalez, S. M. Voronin,Thom’s Problem for the Orbital Analytic Classification of Degen-erate Singular Points of Holomorphic Vector Fields on the Plane,Moscow Mathematical Journal, 12 (4); 2012; pp. 825-862.
Negative refraction tiling billiards
Olga Paris-Romaskevich
Universite de Rennes - 1
A straight line moving through a standard square tiling is as-sociated to a trajectory of a billiard in a square by an unfoldingprocedure. The idea of tiling billiards is to work directly withthe trajectories moving through the tilings. This talk will bedevoted to negative refraction tiling billiards, in which a piece-wise linear trajectory breaks each time it crosses a boundaryof a tile in a way that the refraction coefficient is equal to -1.It happens that even in a very simple periodic tiling by thesame triangle tile, the behaviour of such a system is quiet in-tricate (even though this is a system of zero entropy): sometrajectories of negative refraction tiling billiards approximatefractals (such as Rauzy fractal). I will describe the state ofprogress, as well as the key ideas behind the understandingof these billiards and related foliations, following the work ofP. Baird-Smith, D. Davis, E. Fromm, S. Iyer, P. Hubert andmyself.
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Figure 1: Tiling billiard
On Palis Problem of Embedding ofMorse-Smale Cascades into Flows
Olga Pochinka
Higher School of Economics
The problem of finding of conditions when Morse-Smale cas-cade can be embedded into a topological flow goes back toJ. Palis (1969). He stated some necessary conditions of em-bedding of Morse-Smale cascade 𝑓 : 𝑀𝑛 →𝑀𝑛 (𝑛 ≥ 2) into atopological flow and proved that these conditions are sufficient
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for embedding of Morse-Smal diffeomorphism into a flow for𝑛 = 2. We show that already in the case 𝑛 = 3 they are notsufficient.
The report is devoted to finding a criteria for embeddingof Morse-Smale diffeomorphism of the sphere 𝑆𝑛, 𝑛 ≥ 3, intotopological flow. The results was obtained in collaborationwith V. Grines, E. Gurevich and V. Medvedev [1, 2, 3].
Acknowledgements. This work was supported by the Rus-sian Science Foundation (project 17-11-01041).
REFERENCES
[1] V. Grines, E. Gurevich, V. Medvedev, O. Pochinka, Onthe embedding of Morse-Smale diffeomorphisms on a 3-manifoldin a topological flow, Mat. Sb., 2012, Vol.203, No. 12, 81-104;translation in Sb. Math., 2012, Vol. 203, No. 11-12, 1761 - 1784.
[2] V. Grines, E. Gurevich, O. Pochinka, On embedding Morse-Smale diffeomorphisms on the sphere in topological flows RussianMathematical Surveys (2016), 71 (6):1146
[3] V. Grines, E. Gurevich, O. Pochinka, On embedding of multi-dimensional Morse-Smale diffeomorphisms in topological flows.https://arxiv.org/abs/1806.03468. 2018
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Dmitry Andreevich Gudkov – an outstandingmathematician and the researcher of
Lobachevsky’s biography(to the 100 anniversary since birth)
Grigory Polotovskiy
Lobachevsky State University of Nizhni Novgorod
Life and activity of professor of the Nizhny Novgorod Univer-sity Dmitry Andreevich Gudkov (1918 – 1992) are described.The main mathematical result of D.A. Gudkov (1969) – thetopological classification of real nonsingular curves of degree 6– belong to the known for the difficulty 16th Hilbert problem.At the same time Gudkov formulated a conjecture (“Gud-kov’s congruence”) about topological characteristics of non-singular curve of any even degree. V. I. Arnold’s (1971) andV. A. Rokhlin’s (1972) works on the proof of this congruenceopened the modern stage in development of a subject.
In 1992 D. A. Gudkov published the book“N. I. Lobachevsky. Biography riddles” in which on thebasis of archival documents proved that the Nizhny Novgorodland surveyor S. S. Shebarshin was the father of NikolayLobachevsky and his brothers. Now this statement turnedinto the admitted fact for experts.
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Patterson–Sullivan construction for pointprocesses and the reconstruction for harmonic
functions
Yanqi Qiu
Academy of Mathematics and Systems Science,Chinese Academy of Sciences
In a recent joint work with Alexander Bufetov, we show thatthe classical Patterson-Sullivan construction can be general-ized to the random setting in the theory of point processes.This construction allows us to recover the value of any har-monic function with additional regularity at any point of thedisc from its restriction to a random configuration of the de-terminant point process with the Bergman kernel. Similarresult is then extended to real and complex hyperbolic spacesof higher dimension. Recovering continuous functions by thePatterson-Sullivan construction is also shown to be possible inmore general Gromov hyperbolic spaces.
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Global approximation on Diff𝜔(𝑆1)
Julio Rebelo
Institut de Mathematiques de Toulouse
We will consider finitely generated subgroups 𝐺 of the group of(orientation-preserving) analytic diffeomorphisms of the circle,Diff𝜔(𝑆1). Among finitely generated subgroups of Diff𝜔(𝑆1),groups that are locally non-discrete are extremely common.Here a group 𝐺 as above is said to be locally non-discrete ifthere is a non-trivial sequence of elements {𝑔𝑖} ⊂ 𝐺 whoserestrictions to some fixed, open (non-empty) interval 𝐼 ⊂ 𝑆1
converge to the identity in a suitable topology.
Another interesting class of finitely generated subgroups ofDiff𝜔(𝑆1) is formed by those groups 𝐺 that are globally non-discrete in the sense that 𝐺 contains a non-trivial sequence ofelements 𝑔𝑖 converging to the identity on all of 𝑆1. This latterclass of groups is also very broad, in particular it contains opensets of groups. This class also contains some natural examplesof subgroups of Diff𝜔(𝑆1).
In this talk we will discuss denseness properties for both lo-cally non-discrete and globally non-discrete groups 𝐺 as above.After a quick review of “local approximation” results and theirdynamical applications, we will discuss the “global approxi-mation phenomenon”. This discussion will focus on a recentapproximation result along with some dynamical applicationsthat cannot be derived from the previously known “local ap-proximation” theorems. A typical example in this direction be-ing provided by the problem of existence of regular stationarymeasures. If time permits, I will also comment on some prob-
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lems concerning the borderline between locally non-discretegroups and globally non-discrete ones.
On reduction of singularities forvector fields in dimension 3
Helena Reis
University of Oporto
A classical result due to Seidenberg states that every singu-lar holomorphic foliation on a complex surface can be turnedinto a foliation possessing only elementary singular points bymeans of a finite sequence of (one-point) blow-ups. Here weremind the reader that a singular point 𝑝 is said to be elemen-tary if the foliation ℱ in question has at least one eigenvaluedifferent from zero at 𝑝. However, in dimension 3, the naturalanalogue of Seidenberg theorem no longer holds as shown bySanz and Sancho.
More recently, two major works have been devoted to thisproblem. In [1], Cano, Roche and Spivakovsky have workedout a reduction procedure using (standard) blow-ups. Themain disadvantage of their theorem lies, however, in the factthat some of their final models are quadratic and hence haveall eigenvalues equal to zero. On the other hand, McQuillanand Panazzolo [2] have successfully used weighted blow-ups toobtain a satisfactory desingularization theorem in the categoryof stacks, rather than in usual complex manifolds.
A basic question is how far these theorems can be improvedif we start with a complete vector field on a complex manifold
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of dimension 3, rather than with a general 1-dimensional holo-morphic foliation. In this context of complete vector fields, wewill prove a sharp desingularization theorem. Our proof of thementioned result will naturally require us to revisit the worksof Cano-Roche-Spivakovsky and of McQuillan-Panazzolo ongeneral 1-dimensional foliations. In particular, by buildingon [1], our discussion will also shed some new light on thedesingularization problem for general 1-dimensional foliationon complex manifolds of dimension 3.
REFERENCES
[1] F. Cano, C. Roche & M. Spivakovsky, Reduction of singularitiesof three-dimensional line foliations, Rev. R. Acad. Cienc. ExactasFıs. Nat. Ser. A Math. RACSAM, 108, no 1, (2014), 221–258.
[2] M. McQuillan & D. Panazzolo, Almost etale resolution of folia-tions, J. Differential Geometry, 95, (2013), 279-319.
[3] J. Rebelo & H. Reis, On resolution of 1-dimensional foliations on3-manifolds, available from arXiv (arXiv:1712.10286).
On the classification of germs offoliations in (𝐶𝑛, 0)
Ernesto Rosales-Gonzalez
Instituto de Matematicas,Universidad Nacional Autonoma de Mexico
We consider germs of holomorphic vector fields in (𝐶𝑛, 0), 𝑛 ≥3, with non isolated singularities. We assume that the set ofsingular points forms a submanifold of codimension 2, and thesum of the nonzero eigenvalues of the linearization of the germs
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at each singular point is zero. We give the orbital analyticclassification of generic germs of such type. It happens thatthe formal classification is trivial, and the analytic one hasfunctional moduli, unlike the cases of dicritic and nondicriticgeneric germs in n=2 with isolated and degenerated singularitywhere the formal and analytical classification coincide [2],[3].
Joint work with Ortiz-Bobadilla, L. (UNAM, Mexico);Voronin, S. M. (CSU, Russia).
REFERENCES
[1] Ortiz-Bobadilla, L, Rosales-Gonzalez, E; Voronin, S. M. An-alytic classification of foliations induced by germs of holomorphicvector fields in (C𝑛, 0) with nonisolated singularities. Preprint.
[2] Ortiz-Bobadilla,L. Rosales-Gonzalez, E. Voronin, S. M.,Rigidity theorems for generic holomorphic germs of dicritic fo-liations and vector fields in (C 2 ,0), Mosc. Math. J. 5 (2005), no.1, 171–206.
[3] Ortiz-Bobadilla L., Rosales-Gonzalez E., Voronin, S. M.,Thom’s Problem for Degenerate Singular Points of HolomorphicFoliations in the Plane, Moscow Mathematical Journal, Volume12, N4, 2012. pp. 825-862.
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Erratic behavior for one-dimensional randomwalks in a generic quasi-periodic environment
Maria Saprykina
KTH, Stockholm
Consider a Markov chain on a one-dimensional torus T, wherea moving point jumps from 𝑥 to 𝑥 ± 𝛼 with probabilities𝑝(𝑥) and 1 − 𝑝(𝑥), respectively, for some fixed function 𝑝 ∈𝐶∞(T, (0, 1)) and 𝛼 ∈ R ∖ Q. Such Markov chains are calledrandom walks in a quasi-periodic environment. It was shownby Ya. Sinai that for Diophantine 𝛼 the corresponding randomwalk has an absolutely continuous invariant measure, and thedistribution of any point after 𝑛 steps converges to this mea-sure. Moreover, the Central Limit Theorem with linear driftand variance holds.
In contrast to these results, we show that random walkswith a Liouvillian frequency 𝛼 generically exhibit an erraticstatistical behavior. In particular, for a generic 𝑝, the corre-sponding random walk does not have an absolutely continuousinvariant measure, both drift and variance exhibit wild oscil-lations (being logarithmic at some times and almost linear atother times), Central Limit Theorem does not hold.
These results are obtained in a joint work with DmitryDolgopyat and Bassam Fayad.
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Sparkling saddle loops of vector fields on surfaces
Ivan Shilin
Higher School of Economics, Moscow, Russia
A vector field on a two-dimensional surface different from thesphere and projective plane can have a dissipative saddle witha saddle loop that attracts the “free” unstable separatrix ofthis saddle. When the loop is unfolded in a generic one-parameter family, new saddle loops appear for parameter val-ues arbitrarily close to zero. These are sparkling saddle con-nections which are loops – hence the name.
When we study this as a semi-local bifurcation, it turns outthat in the orientable case the closure of the set of parametersfor which the field has a saddle loop is a Cantor set. By modi-fying the field outside a neighborhood of the unstable manifoldof our saddle, it is possible to add a fixed but arbitrary numberof bifurcation points into each interval of the complement ofthe Cantor set, thus obtaining countably many nonequivalentbifurcation diagrams for local one-parameter families of vectorfields on any surface other than the sphere, Klein bottle, andprojective plane.
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Cohomological equations for linear involutions
Sasha Skripchenko
Higher School of Economics, SkolTech
Famous Roth theorem about diophantine approximationsstates that a given algebraic number may not have too manyrational number approximations, that are “very good”. Moreprecisely, Roth first defined a class of numbers that are notvery easy to approximate by rationals (they are called Rothnumbers) and then showed that almost all algebraic irrationalsare of Roth type, and that they form a set of a full measurewhich is invariant under the natural action of the modulargroup SL(2,Z).
In addition to their interesting arithmetical properties,Roth type irrationals appear in a study of the cohomologi-cal equation associated with a rotation 𝑅𝑎 : 𝑅𝑎(𝑥) = 𝑥 + 𝑎of the circle T = R/Z: 𝑎 is of Roth type if and only if for all𝑟, 𝑠 : 𝑟 > 𝑠 + 1 > 1 and for all functions Φ of class 𝐶𝑟 on Twith zero mean there exists a unique function Ψ ∈ 𝐶𝑠(T) withzero mean such that
Ψ − Ψ ∘𝑅𝑎 = Φ.
In 2005 Marmi, Moussa and Yoccoz established an ana-logue of Roth theorem for interval exchange transformations(IETs). In particular, they defined the notion of Roth typeIETs and proved existence of the solution of cohomologicalequation for this class; they also showed that IET of Rothtype form a full measure set in the parameter space of IETs.
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In a very fresh joint work with Erwan Lanneau and StefanoMarmi we get a certain generalization of this result for linearinvolutions that can be considered as a natural extension ofIETs to non-orientable case.
Bifurcations of families of vector fieldson the two-dimensional sphere
Nikita Solodovnikov
Higher School of Economics
There exists pair of weakly topologically equivalent vectorfields on the two-sphere with a parabolic cycle such thattheir generic one-parameter unfoldings are not equivalent ([1]).Classification of generic one-parameter unfoldings of vectorfields with parabolic cycles is given. Any pair of weaklytopologically equivalent degenerate vector fields of codimen-sion 1 without parabolic cycle can have only equivalent one-parameter families as unfoldings.
An overview of bifurcations of generic one-parameter fam-ilies and proof of the result stated above is provided.
The author is supported by RFBR project 16-01-00748-a
REFERENCES
[1] N. Goncharuk, Y. Ilyashenko, N. Solodovnikov. Global bifurca-tions in generic one-parameter families with a parabolic cycle on𝑆2. arXiv:1707.09779
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Holomorphic normal form of nonlinearperturbations of nilpotent vector fields
Laurent Stolovitch
CNRS, Laboratoire J.-A. Dieudonne, U.M.R. 6621,Universite Cote d’Azur
We consider germs of holomorphic vector fields at a fixed pointhaving a nilpotent linear part at that point, in dimension𝑛 ≥ 3. Based on Belitskii’s work, we know that such a vectorfield is formally conjugate to a (formal) normal form. We givea condition on that normal form which ensure that the nor-malizing transformation is holomorphic at the fixed point. Weshall show that this sufficient condition is a nilpotent version ofBruno’s condition (A). In dimension 2, no condition is requiredsince, according to Strozyna-Zo ladek, each such germ is holo-morphically conjugate to a Takens normal form. Our proof isbased on Newton’s method and sl2(C)-representations.
REFERENCES
[Bel79] G. R. Belitskii. Invariant normal forms of formal series. Funct.Anal. Appl., 13(1):46–47, 1979.
[Bru72] A.D. Bruno. Analytical form of differential equations. Trans.Mosc. Math. Soc, 25,131-288(1971); 26,199-239(1972), 1971-1972.
[LS10] E. Lombardi and L. Stolovitch. Normal forms of analyticperturbations of quasihomogeneous vector fields: Rigidity, in-variant analytic sets and exponentially small approximation.Ann. Scient. Ec. Norm. Sup., pages 659–718, 2010.
[SV16] L. Stolovitch and F. Verstringe. Holomorphic normalforms of perturbations of nilpotent vector fields. Regular andChaotic Dynamics, 21(4):410–436, 2016.
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[SZ02] E. Strozyna and H. Zo ladek. The analytic and formal normalform for the nilpotent singularity. J. Differential Equations,179(2):479–537, 2002.
[SZ08] E. Strozyna and H. Zo ladek. Multidimensional formal Tak-ens normal form. Bull. Belg. Math. Soc. Simon Stevin, 15(5,Dynamics in perturbations):927–934, 2008.
On a heavy quantum particle
Dmitry Treschev
Steklov Mathematical Institute of Russian Academy of Sciences
We consider Schrodinger equation for a particle on a flat 𝑛-torus in a bounded potential, depending on time. Mass ofthe particle equals 1/𝜇2, where 𝜇 is a small parameter. Weshow that the Sobolev 𝐻𝜈-norms of the wave function growapproximately as 𝑡𝜈 on the time interval 𝑡 ∈ [−𝑡*, 𝑡*], where 𝑡*is slightly less than 𝑂(1/𝜇).
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On classification of Morse-Smale dynamicalsystems with few non-wandering points
Evgeny Zhuzhoma
National Research University Higher School of Economics
We consider Morse-Smale flows and diffeomorphisms with thenon-wandering set consisting of three fixed points (two nodesand a saddle). We study the topological structure of support-ing manifolds. The question of the topological classificationis considered also. The results obtained in collaboration withV. S. Medvedev.
This work was supported by Russian Scientific Found(RNF), project 17-11-01041.
Flat and hyperbolic enumerative geometry
Anton Zorich
Institute of Mathematics of Jussieu and CAS, Skoltech
This talk (based on joint work with V. Delecroix, E. Goujardand P. Zograf) bridges certain aspects of flat and hyperbolicenumerative geometry.
We derive from the Kontsevich’s count of metric ribbongraphs a formula for the Masur–Veech volume of the modulispace 𝒬𝑔,𝑛 of quadratic differentials in terms of intersectionnumbers of 𝜓-classes.
We show that Mirzakhani’s frequencies of simple closedhyperbolic geodesics of different combinatorial types coincidewith the frequencies of the corresponding square-tiled surfaces.
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Using our recent equidistribution results for square-tiledsurfaces of fixed combinatorial type we derive applications forthe count of meanders on surfaces of arbitrary genus.
We conclude with several conjectures concerning the shapeof a typical multicurve on a surface of large genus and theshape of a typical square-tiled surface of large genus.
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Posters
Energy function for 3-diffeomorphisms withone-dimensional surface attractor and repeller
Marina Barinova
Higher School of Economics, Nizhny Novgorod Campus
The Lyapunov function was introduced by A. M. Lyapunovto study the stability of equilibrium states of differential equa-tions. A smooth Lyapunov function in which the set of criticalpoints coincides with the chain-recurrent set of the system iscalled the energy function. For example, such functions al-ways exist for flows, whereas examples of cascades are knownwhich do not have an energy function. So there always existsa Morse-Lyapunov function for arbitrary gradient-like Morse-Smale diffeomorphisms, but, in general, there is no Morse en-ergy function. The first example of such diffeomorphism wasbuilt on the 3-sphere by Picston in 1977 based on the wild arcof Artin-Fox. The Pikston construction allows generalizationto any dimensions, thereby giving in the general case a negativeanswer to the question of the existence of an energy functionin systems with regular dynamics. The Pikston construction
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allows generalization to any dimensions, thereby giving in thegeneral case a negative answer to the question of the existenceof an energy function in systems with regular dynamics. Themore surprising is the fact that there exists an energy func-tion for cascades with chaotic dynamics, discovered in 2015 byV. Z. Grines, M. K. Barinova and O. V. Pochinka. The Pikstonconstruction allows generalization to any dimensions, therebygiving in the general case a negative answer to the questionof the existence of an energy function in systems with regulardynamics. The more surprising is the fact that there exists anenergy function for cascades with chaotic dynamics, discoveredin 2015 by V. Z. Grines, M. K. Barinova and O. V. Pochinka.They established the existence of the energy function for sur-face omega-stable diffeomorphisms with one-dimensional ba-sis sets and for rough three-dimensional diffeomorphisms withsurface basis sets. This report is a continuation of this topic.We consider Ω-stable 3-diffeomorphisms, the nonwandering setof which consists of exactly one surface attractor and one sur-face repeller. We build an example of such a diffeomorphismand prove the existence of the energy function of such diffeo-morphisms.
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Central Limit Theorem for Laguerre EnsemblesUnder Double Scaling
Sergey Berezin 1 and Alexander Bufetov 2
1 Peter the Great St. Petersburg Polytechnic University;2 CNRS, Steklov IITP RAS
[email protected]; [email protected]
The Central Limit Theorem is obtained for additive function-als of Laguerre random matrices in a double-scaling limit asboth the matrix size and the scaling parameter of the modelgo to infinity. The Riemann–Hilbert approach is used to findthe asymptotic behaviour of the exponential moments.
Difference between smooth and formalclassifications of Morse transitions of implicit
ODE
Ilya Bogaevsky
Lomonosov Moscow State University, Scientific Research Institutefor System Analysis
A topological classification of Morse transitions in families ofimplicit ordinary differential equations
𝐹 (𝑥, 𝑦, 𝑦′, 𝜀) = 0
depending on a parameter 𝜀 ∈ R is obtained in [1], consists offour types, and does not contain continuous invariants. Thephase portrait of one of them — a disk appearing from nothing— is shown below.
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𝜀 < 0 𝜀 = 0 𝜀 > 0
The shown transition has the topological normal form
𝑦′2
+ 𝑥2 + 𝑦2 = 𝜀.
Its asymptotic (formal) normal form
𝑦′2
+ 𝑥2 + 𝑦2 + 𝑐(𝜀)𝑦3 = 𝜀
with one functional invariant 𝑐2 is found in [2].
It turns out that its smooth normal form is more com-plicated and contains at least one more invariant — a flatfunction of two variables which can not be killed by smoothchanges of the parameter 𝜀 and the variables 𝑥, 𝑦 dependingon the parameter.
The work is partially supported by RFBR-16-01-00766.
REFERENCES
[1] J. W. Bruce, G. J. Fletcher, and F. Tari. Bifurcations of im-plicit differential equations. Proceedings of the Royal Society ofEdinburgh: Section A Mathematics, 130(3):485–506, 2000.
[2] I. A. Bogaevsky. Implicit ordinary differential equations: bifurca-tions and sharpening of equivalence. Izv. Math., 78(6):1063–1078,2014.
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On the circumcenters of triangular orbits inelliptic billiard and complex reflexion law
Corentin Fierobe
ENS Lyon
Here is a little problem: take an ellipse and look at the tri-angular billard orbits inscribed in it. What can we say aboutthe locus of their circumcenters? We prove that it is an ellipse[Fie18]. The method of proof is based on a complex reflexionlaw introduced by Alexey Glutsyuk [Glu14b, Glu14a, Glu17]in order to study Ivrii’s conjecture on periodic orbits togetherwith its analogues for pseudo-billiards and complex billiards.Olga Romaskevich [Rom14] proved that the incenters of thetriangular orbits form an ellipse, by using the above-mentionedcomplex reflection law. Our proof follows a similar scheme, asin [Rom14].
REFERENCES
[Fie18] C. Fierobe. On the circumcenters of triangular orbits in ellipticbillards. ArXiv:1807.11903, July 2018.
[Glu14a] A. Glutsyuk. On odd-periodic orbits in complex planar bil-liards. Journal of Dynamical and Control Systems, 20:293–306, 2014.
[Glu14b] A. Glutsyuk. On quadrilateral orbits in complex algebraicplanar billiards. Moscow Mathematical Journal, 14:239–289,2014.
[Glu17] A. Glutsyuk. On 4-reflective complex analytic billiards. Journalof Geometric Analysis, 27:183–238, 2017.
[Rom14] O. Romaskevich. On the incenters of triangular orbits in el-liptic billiard. Enseign. Math. 60, no. 3-4, 247-255, 2014.
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Evolution of moments of isotropic Brownianstochastic flows
Vladimir Fomichov
Institute of Mathematics, National Academy of Sciences of Ukraine
We consider the following stochastic integral equation:
𝑥(𝑢, 𝑡) = 𝑢+
𝑡∫0
∫R
𝜙(𝑥(𝑢, 𝑠) − 𝑞)𝑊 (𝑑𝑞, 𝑑𝑠), 𝑡 > 0, 𝑢 ∈ R,
where 𝑊 is the Wiener sheet on R×R+ and the function 𝜙 ∈𝐶∞0 (R) is non-negative and such that 𝜙(𝑞) = 𝜙(−𝑞), 𝑞 ∈ R,
and∫R 𝜙
2(𝑞) 𝑑𝑞 = 1. It is known (see [1]) that under theseconditions on the function 𝜙 this equation has a unique strongsolution for every 𝑢 ∈ R, and the corresponding family ofrandom mappings 𝑥(·, 𝑡) : R → R, 𝑡 > 0, defines an isotropicBrownian stochastic flow of 𝐶∞-diffeomorphisms (see [3]).
For this stochastic flow we establish the precise asymptoticbehaviour of all mixed moments and of all moments of theinterparticle distance.
These results can be easily extended to the general caseof one-dimensional isotropic Brownian stochastic flows withsome non-restrictive conditions imposed on their covariancefunction.
REFERENCES
[1] Dorogovtsev, A. A. One Brownian stochastic flow // Theory ofStochastic Processes 10(26):3-4 (2004) 21–25.
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[2] Fomichov, V. V. Evolution of moments of isotropic Brownianstochastic flows // Theory of Stochastic Processes 20(36):1 (2015)14–27.
[3] Kunita, H. Stochastic flows and stochastic differential equations. —Cambridge University Press, 1990. — 346 p.
Topological conjugacy of Ω-stable flows withoutlimit cycles on surfaces
Vladislav Kruglov
NRU HSE – Nizhnij Novgorod
Recall that topological conjugacy for flows means existence ofa homeomorphisms sending trajectories of one flow into tra-jectories of another preserving orientation and time of moving.For continuous systems (flows) conjugacy differs of topologicalequivalence, which does not obvious preserve time of moving.
In this work we speak about Ω-stable dynamical systemson closed surfaces. The non-wandering set of such systemsconsists of a finite number of hyperbolic periodic orbits andfixed points which does not construct cycles. Considered sys-tems are not structural stable in a general case because ofopportunity of a heteroclinic tangency. In case of flows suchtangency means coincidence of two separatrices of differentsaddle points; such separatrix is called connection.
So, any neighbourhood of an Ω-stable system in the spaceof dynamical systems may consist infinitely many pairwise dif-ferent classes of topological conjugacy. The parameters de-scribing classes in some such neighbourhood are called moduliof topological conjugacy.
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In 1978 J. Palis [1] invented a continuum of topologicallynon-conjugated dynamical systems in a neighbourhood of asystem with a heteroclinic tangency on a surface. He de-scribed a neighbourhood of a heteroclinic tangency trajectoryand showed that conjugacy needs coincidence of moduli. Amodulus depends on eigenvalues of the saddles whose separa-trices tangent.
Later W. de Melo and S. Van Strien in [2] in 1987 charac-terised a class of surface diffeomorphisms with a finite numberof moduli. They found that a chain of saddles taking part ina heteroclinic tangency can not be more than two. But thereis no need in such property in case of flows.
In this work we show that a number of moduli for flowswithout limit cycles is finite and coincides with a number ofconnections. A complete topological invariant for such systemsis four-colour graph carrying information about dividing thephase space into domains with similar trajectories behaviour(cells). Edges of this graph are equipped with analytical pa-rameters – moduli, connected with saddle connections. Suchgraph generilize the Oshemkov-Sharko three-colour graph [3]for Morse-Smale flows on surfacesand four-colour graph from[4], which is the invariant of topological equivalence of Ω-stableflows without limit cycles on surfaces.
Acknowledgements. Author thanks Olga Pochinka for irre-placeable help. The research was funded by RFBR accordingto the research project No 18-31-00022 mol a.
REFERENCES
[1] J. Palis, A Differentiable Invariant of Topological Conjugacies andModuli of Stability, Asterisque, 51 (1978), 335-346.
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[2] W. de Melo, S. J. van Strien, Diffeomorphisms on surfaces with afinite number of moduli, Ergodic Theory and Dynamical Systems,7 (1987), 415-462.
[3] A. A. Oshemkov, V. V. Sharko, On classification of Morse-Smaleflows on 2-dimensional manifolds (in Russ.), Matematicheskiysbornik, 189:8 (1998), 93-140.
[4] V. E. Kruglov, D. S. Malyshev, O. V. Pochinka, Multicolourgraph as a complete topological invariant for Ω-stable flows with-out periodic trajectories on surfaces (in Russ.), Matematicheskiysbornik, 209:1 (2018), 100-126.
Classification of one-dimensional attractors bymeans of Pseudo-Anosov homeomorphisms
Evgeny Kurenkov
Higher School of Economic, Nizhny Novgorod
In the present paper axiom 𝐴 diffeomorphisms of a closed 2-manifold 𝑀2 of genus 𝑝 ≥ 2 whose nonwandering sets containspaciously situated basic set Λ are considered. It is shownthat the problem of topological classification of such diffeo-morphisms is closely related to the problem of topological clas-sification of Pseudo-Anosov homeomorphisms on surfaces.
Definition 1. One-dimensional basic set Λ of an axiom 𝐴 dif-feomorphism 𝑓 : 𝑀2 →𝑀2 is called if perfect its complement𝑀2 ∖ Λ consists of a finite union of open disks.
In the paper [1] it was announced that for any Pseudo-Anosov homeomorphisms of a surface it is possible to constructa homotopic structurally stable diffeomorphism with a perfect
49
basic set by means of a surgery operation. However, the prob-lem of topological classification of such diffeomorphisms wasnot considered there.
Theorem 1. Let 𝑓 : 𝑀2 → 𝑀2 be a diffeomeorphism of anorientable 2-manifold 𝑀2 of genus 𝑝 ≥ 2 with a perfect spa-ciously situated basic set Λ, then there exist a continuous mapℎ : 𝑀2 →𝑀2 homotopic to the identity and a Pseudo-Anosovhomeomorphism 𝑃𝐴𝑓 : 𝑀2 →𝑀2 such that the following dia-gram commutes ℎ ∘ 𝑓 = 𝑃𝐴𝑓 ∘ ℎ.
We say that a periodic point point 𝑝 is tagged if the setℎ−1(𝑝)∩Λ consist of exactly two distinct points. Let 𝐵 denotethe set of all tagged points. The main result is the followingtheorem.
Theorem 2. Let 𝑓 and 𝑓 ′ be 𝐴-diffeomorphisms of a closedorientable 2-manifold 𝑀2 of genus 𝑝 ≥ 2, with perfect spa-ciously situated attractors Λ and Λ′ correspondingly. Thenthere exists a homeomorphism 𝑔 : 𝑀2 →𝑀2 such that the fol-lowing diagram commutes 𝑓 ′|Λ′ = 𝑔 ∘ 𝑓 ∘ 𝑔−1|Λ′ if and onlyif there exists a homeomorphism ℎ : 𝑀2 → 𝑀2 such that𝑃𝑓 ′ = ℎ ∘ 𝑃𝑓 ∘ ℎ−1 and ℎ(𝐵) = 𝐵′.
Acknowledgments. Research is done with financial supportof Russian Science Foundation (project 17-11-01041)
REFERENCES
[1] Zhirov, A. Yu., Plykin, R. V. (1995). Correspondence between one-dimensional hyperbolic attractors of surfaces diffeomorphism andgeneralized Pseudo-Anosov diffeomorphisms. Mathematical Notes,58(1), 149-152.
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The space of cubic Newton maps with persistentparabolics
Khudoyor Mamayusupov
National Research University Higher School of EconomicsFaculty of Mathematics, Usacheva 6, Moscow, Russia
We study the space of cubic Newton maps for entire maps thattake the form (𝑧2 + 𝑐)𝑒𝑧 and the space is parametrized by asingle complex parameter 𝑐 = 0. These Newton maps are ra-tional, 𝑧 ↦→ 𝑓𝑐(𝑧) = 𝑧 − 𝑧2+𝑐
𝑧2+2𝑧+𝑐, with a persistent parabolic
fixed point at ∞ and two superattracting fixed points in C,and all have a connected Julia set [M1]. We show that sta-ble components, the components of cubic Newton maps forwhich the free critical point belongs to attracting basins andthe basin of the parabolic fixed point at ∞, are topologicaldisks. When the free critical point belongs to the basin of ∞,we consider quasiconformal conjugacy classes in a correspond-ing stable component, which is a topological disk comprised ofhalf-closed strips, which is a bi-infinite cylinder obtained ex-plicitly as a class of quasiconformal deformations of some 𝑓𝑐 inthis space [BMMS]. Each deformation class contains a uniquesuch 𝑓𝑐, which is conformally rigid and we call these maps Post-Critically Minimal and PostCritically Non-Minimal accordingto the number of iterates its free critical point takes to reachthe other non-free critical point in the parabolic basin [M2].Furthermore, by means of quasiconformal surgery we constructa canonical bijection from the hyperbolic components of thespace of standard cubic Newton maps of polynomials to thestable components of this one-parameter space [M1, M2].
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REFERENCES
[BMMS] K. Bogdanov, K. Mamayusupov, S. Mukherjee and D.Schleicher, Antiholomorphic perturbations of WeierstrassZeta functions and Green’s function on tori, Nonlinearity. Vol.30. No. 8. pp 3241–3254. (2017)
[M1] K. Mamayusupov, Newton maps of complex exponential functionsand parabolic surgery, Fundamenta Mathematicae, Vol. 241. No.3. P. 265-290. (2018)
[M2] K. Mamayusupov A characterization of postcritically minimalNewton maps of complex exponential functions, Ergodic Theoryand Dynamical Systems. http://doi.org/10.1017/etds.2017.137(2018)
Modulus of continuity away from zero forcorrelation measures in substitution tiling
systems
Juan Marshall
Universite Aix-Marseille
A historically powerful tool in ergodic theory is the spectralanalysis of Koopman operators arising from a dynamical sys-tem. The spectral theorem identify these operators with theso called spectral measures, making them an important ob-ject of study. As an example of this, you can read the excel-lent book of Queffelec [Que09], where the author characterizessome spectral measures in the case of one-dimensional sym-bolic substitutions, as a combination of Dirac’s deltas.
More recently, Bufetov and Solomyak studied these mea-sures in tilings systems of the real line arising from substitu-
52
tions [BS14], but with the difference of being focused in con-tinuous spectra, i.e., the weakly-mixing case. They were in-spired by some previous works of Hof [Hof97]. Riesz productsare used to characterize spectral measures, and some meth-ods from Diophantine approximation theory to estimate theseproducts.
Some of those results are generalized to some Cantor sys-tems in a more recent paper of the same authors. In otherdirection, one result of [BS14] is generalized to substitutiontiling systems of R𝑑: in [Emm17] the modulus of continuity ofspectral measures around zero is characterized by some Radonmeasure. In this way, the aim of the actual paper is the studyof decay rates of the spectral measures of substitution tilingsystems of R𝑑, but at points different from zero. The nexttheorem is the main result of this paper and is a natural gen-eralization of Theorem 5.1 in [BS14]:
Theorem 1. Let 𝜁 a tile-substitution with expansion 𝐿 overthe alphabet 𝐴 = {𝑇1, . . . , 𝑇𝑚} and 𝑓 = 1X𝑇𝑖
(characteristic
function of the set of tilings covering 0 ∈ R𝑑 with the tile𝑇𝑖). Suppose 𝐿 is diagonalizable over C, its eigenvalues areall Galois conjugates and there exists one conjugate outsidethe unit disk and not in the spectrum of 𝐿. Consider the set ofΞ the set of 𝜔’s such that 𝑣 /∈ ⟨{𝜔}⟩⊥ and lim inf 𝑙≥0 |𝜔 ·𝐿𝑙𝑣| >1/2, for some 𝑣 ∈ R𝑑 depending on 𝜁. Then, for 𝜀 > 0 thereexist constants 𝐶𝜀, 𝛾 > 0 such that
𝜎𝑓 ([𝜔1−𝑟, 𝜔1+𝑟]×· · ·×[𝜔𝑑−𝑟, 𝜔𝑑+𝑟]) ≤ 𝐶𝜀 log(1/𝑟)−𝛾 , 𝑟 ↘ 0.
for 𝜔’s in Ξ such that |∑𝑚1
𝑖=1 𝜂𝑖1𝜔 ·𝑢𝑖1| ∈ [𝜀, 1/𝜀], where {𝑢𝑖1}
𝑚1𝑖=1
are the eigenvectors of 𝐿 associated to the greater (in modulus)eigenvalue and 𝑚1 its multiplicity.
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REFERENCES
[BS14] Alexander I. Bufetov and Boris Solomyak, On the mod-ulus of continuity for spectral measures in substitution dy-namics, Advances in Mathematics 260 (2014), 84–129.
[Emm17] Jordan. Emme, Spectral measure at zero for self-similar tilings,Mosc. Math. J. 17 (2017), no. 1, 35–49.
[Hof97] A. Hof, On scaling in relation to singular spectra, Communica-tions in Mathematical Physics 184 (1997), no. 3, 567–577.
[Que09] Martine Queffelec, Substitution dynamical systems - spec-tral analysis, 2nd ed., Springer, Berlin, 2009.
Invariant surfaces for Toric type foliations indimension 3
Beatriz Molina-Samper
University of Valladolid (Spain)
Few years ago, Rene Thom asked the following question: Isthere always an invariant surface for a given codimensionone foliation on (C𝑛, 0)? C. Camacho and P. Sad proved,in 1982, that any foliation of (C2, 0) has an invariant curve.For greater dimension, there are results of Cano-Cerveau andCano-Mattei, that give a positive answer to Thom’s questionfor non-dicritical foliations. However, when the existence ofdicritical components of the exceptional divisor is allowed,Jouanolou gives a collection of conic foliations on (C3, 0) with-out invariant surface.
What we present in this poster, is a result of existence ofinvariant surface for, dicritical or not, toric type foliations on(C3, 0). By toric type we mean the existence of a combinatorial
54
process of reduction of singularities. This is a slightly strongercondition than being a Newton non-degenerate foliation (inthe classical sense in terms of Newton polyhedra). In orderto prove this three-dimensional result of local nature, we stateand prove the following global result in dimension two: “Theisolated invariant curves of toric type foliations on compacttoric surfaces are closed curves”.
On displacement of viscous liquid in a quasi 1Dsystem
Grigorii Monakov
Saint-Petersburg State University
An explicitly solvable quasi 1D model of oil displacement isstudied. The problem of recovering of the reservoir geometryis solved by means of a fixed point algorithm. The stability ofsolution is studied in various functional classes.
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On Number Rigidity for Pfaffian point processes
Pavel Nikitin
St. Petersburg Department of V.A.Steklov Institute ofMathematics, St. Petersburg State University
Our first result states that the orthogonal and symplecticBessel processes are rigid in the sense of Ghosh and Peres.Our argument in the Bessel case proceeds by an estimate ofthe variance of additive statistics in the spirit of Ghosh andPeres. Second, a sufficient condition for number rigidity ofstationary Pfaffian processes, relying on the Kolmogorov cri-terion for interpolation of stationary processes and applicable,in particular, to pfaffian sine-processes, is given in terms of theasymptotics of the spectral measure for additive statistics.
Joint work with A. Bufetov and Y. Qiu.
On periodic data of diffeomorphisms with onesaddle orbit
Elena Nozdrinova
NRU HSE – Nizhnij Novgorod
In the study of discrete dynamical systems, i.e., the study oforbits of self-mapsf defined on a given compact manifold, theperiodic behavior plays an important role. In the last fortyyears there was a growing number of results showing that cer-tain simple hypotheses force qualitative and quantitative prop-erties (like the set of periods) of a system. One of the best-known results is the title of the paper “Period three implies
56
chaos for the interval continuous self-maps” [5]. The effect de-scribed there was discovered by O. M. Sarkovs’kiı in [7]. Themost useful tools for proving the existence of fixed points or,more generally, of periodic points for a continuous self-map fof a compact manifold is the Lefschetz fixed point theorem andits improvements (see, for instance [2] and [3]). The Lefschetzzeta-function simplifies the study of the periodic points of f.This is a generating function for all the Lefschetz numbers ofall iterates of f. The periodic data of diffeomorphisms with reg-ular dynamics on surfaces were studied by means zeta-functionin a series of already classical works by such authors as PaulR. Blanchard, John M. Franks, Rufus Bowen, Steve Batter-son, John Smillie, William H. Jaco, Peter B. Shalen, CarolynC. Narasimhan, and others. A description of periodic dataof gradient-like diffeomorphisms of surfaces was given in [1]by means of classification of periodic surface transformationsobtained by Jakob Nielsen [6]. In [4], the authors show thatthe study of periodic data of arbitrary Morse–Smale diffeo-morphisms on surfaces is reduced by filtration to the problemof computing periodic data of diffeomorphisms with a uniquesaddle periodic orbit. Polar diffeomorphisms of the surfaceare considered, that is, iffeomorphisms having a single sinkand a single source periodic orbit. A classic example of such adiffeomorphism is the sink-source diffeomorphism, which hasno saddle points and exists only on a two-dimensional sphere.However, the addition of even a single saddle orbit greatlyexpands the class of polar diffeomorphisms on surfaces. Allpossible types of periodic data for such polar diffeomorphismshave been established, and it is shown that the saddle orbitalways has a negative orientation type. It is proved that ev-
57
ery orientable surface admits a Morse–Smale diffeomorphismwith one saddle orbit that preserves orientation. It is shownthat these diffeomorphisms have exactly three nodal orbits. Inaddition, all possible types of periodic data for such diffeomor-phisms are established.
Acknowledgements. Author thanks Olga Pochinka forirreplaceable help. The research of periodic data for dif-feomorphisms with a single saddle orbit was carried out withthe support of grant RNF 17-11-01041.
REFERENCES
[1] A. N. Bezdenezhnykh and V. Z. Grines, Realization of gradient-like diffeomorphisms of twodimensional manifolds (Russian) inDifferential and Integral Equations (Russian). Ed. N. F.Otrokov.Gorki: Gor’kovskii Gos. Univ., 1985. 33–37, 124–125. Trans. Sel.Math. Sov. 11 (1985), no. 1, 19–23.
[2] Robert F. Brown, The Lefschetz Fixed Point Theorem. Glenview,Ill.-London: Scott, Foresman and Co., 1971.
[3] John M. Franks, Homology and Dynamical Systems. CBMS Re-gional Conference Series in Mathematics, 49. Providence, R. I.:American Mathematical Society, 1982.
[4] V. Z. Grines, O. V. Pochinka, and S. van Strien, A completetopological classification of Morese–Smale diffeomorphisms onsurfaces: A kind of kneading theory in dimension two.
[5] T. Y. Li and James A. Yorke, Period three implies chaos,Amer. Math. Monthly 82 (1975), no. 10, 985–992. Available atarXiv:1712.02230 [Math DS] December 6, 2017.
[6] Jakob Nielsen, Die Struktur periodischer Transformationen vonF1achen. Phys. Meddelerser 15, no. 1, (1937), 1–77.
[7] O. M. Sarkovs’kiı, Co-existence of cycles of a continuous mappingof the line into itself (Russian), Ukrain. Mat. Z. 16 (1964), 61–71.
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Normal forms of infinite index transversalsystems with one consrtraint
Natalia Paziy
Chelyabinsk State University
We consider locally analytic constrained systems with one con-straint of the form
0 = 𝐹 (𝑥, 𝑢, 𝑦)
�� = 𝑣1(𝑥, 𝑢, 𝑦), (𝑥, 𝑢, 𝑦) ∈ (C× C× C𝑛, 0) (1)
�� = 𝑣2(𝑥, 𝑢, 𝑦)
such that 𝐹 (0) = 0, 𝐹 ′(0) = 0, and the vector field 𝑣 = (𝑣1, 𝑣2)does not vanish at the origin.
Systems of the form (1) of finite index, that is, the case
when at least one partial derivative 𝜕𝑘𝐹𝜕𝑥𝑘 (0) does not vanish,
were earlier considered in [1]. We say that the index of (1) is
infinite if 𝜕𝑘𝐹𝜕𝑥𝑘 (0) for all 𝑘 ≥ 0. In this case, a normal form of
(1) can be obtained from the following theorem, which is ourmain result:
Theorem 1. A generic system (1) of infinite index is locallyanalytically equivalent to the system
0 = 𝑢+ 𝑦1𝑥+ 𝑦2𝑥2 + . . .+ 𝑦𝑛𝑥
𝑛 + 𝑥𝑛+1𝐴(𝑥, ′𝑦), �� = 1, �� = 0,
where ′𝑦 = (𝑦2, . . . , 𝑦𝑛), 𝐴(𝑥, 0) ≡ 0 for 𝑛 ≥ 2, and 𝐴 ≡ 0 for𝑛 = 1.
The proof of Theorem 1 is based on the results of [2, 3].The research was supported by RFBR 16-01-00766.
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REFERENCES
[1] N.D. Pazii, Locally analytic classification of equations of Sobolevtype. Ph.D. thesis, Chelyabinsk State University, 1999.
[2] L. Ortiz-Bobadilla, N. Paziy, E. Rosales-Gonsalez, Normalforms of generic transversal systems of infinite index, InternationalConference, dedicated to I.G Petrovskii, (XXIII Joint Session ofPetrovskii Seminar and Moscow Math. Soc. ). Book of abstracts,84–85 (2011).
[3] V. I. Savel’ev, Zero-type embeddings of the sphere into complexsurfaces. Mosc. Univ. Math. Bull. 37, No. 4, 34–39 (1982)
Periodic solutions of one-dimensional parabolicproblems with hysteresis on the boundary
Polina Perstneva
Chebyshev Laboratory, Saint-Petersburg State University, Russia
We study problems which one can interpret as a temperatureregulation in a domain via heating (and cooling) elements onthe boundary. The regime of the heating elements on theboundary is based on the registration of thermal sensors in-side the domain and obeys a hysteresis law. Long-time be-haviour of such systems, in particular, existence and stabilityof periodic solutions, was studied in [1], [2] and [3] by Fouriermethod and Poincare map contraction property. It was es-tablished that the biggest difficulty in study of such systemsoccurs whenever the gap between the switching temperaturesof the hysteresis operator 𝛽 − 𝛼 is too small. For example, in[3] coexistence of several periodic solutions with different sta-bility properties is proved to be possible when 𝛽 − 𝛼 is small
60
enough. We considered one-dimensional system with the meantemperature values 𝑢(0, 𝑡), 𝑢(𝜋2 ) and 𝑢(𝜋, 𝑡) (temperature val-ues at the beginning, the center and 2 the end of the segmentrespectively) and proved that, in contrast to multi- dimen-sional case, for any 𝛽 − 𝛼 existence, uniqueness and stabilityof periodic solutions take place.
REFERENCES
[1] P. Gurevich, W. Jager, A. Skubachevskii, On periodicity of so-lutions for thermocontrol problems with hysteresis-type switches,SIAM J. Math. Anal., 41, No. 2 (2009).
[2] P. Gurevich, Periodic solutions of parabolic problems with hysteresison the boundary,Discrete and Continuous Dynamical Systems, 29,No. 3 (2011).
[3] P. Gurevich, S. Tikhomirov, Symmetric periodic solutions ofparabolic problems with hysteresis, Journal of Dynamics and Dif-ferential Equations, 23 (2011).
Yu. S. Ilyashenko – from local to global
Sergei Pilyugin
St. Petersburg State University, St. Petersburg, Russia
We discuss some of Yu. S. Ilyashenko results and the individu-ality of his mathematical style.
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Dynamics of geodesics of meromorphicconnections on compact Riemann surfaces
Karim Rakhimov
University of Pisa
In this work we have studied relation among three ob-jects: quadratic differentials (𝒬𝒟), meromorphic flat surfaces(ℳℱ𝒮) and meromorphic connections (ℳ𝒞) on Riemann sur-faces. First of all, we have shown that there exists a subset𝒜 ( ℳ𝒞 such that 𝒬𝒟 ∼= 𝒜. We have shown that if (𝑆, 𝑔)is a meromorphic flat surface (𝑆 is a Riemann surface withpunctures and 𝑔 is a flat metric outside the punctures) thenthere exists a meromorphic connection such that 𝑔 will be themetric adapted to the meromorphic connection. In particular,ℳℱ𝒮 ( ℳ𝒞.
In this work we have defined a notion of argument for ameromorphic connection and by using it we have proved someproperties of self-intersecting geodesics. As a consequence, wehave seen that the imaginary part of residues of poles of ameromorphic connection does not significantly affect dynami-cal properties of the meromorphic connection.
We also studied local behavior of geodesics of a meromor-phic connection in a neighbourhood of Fuchsian poles. Wehave seen that if the omega-limit set of a geodesic of a mero-morphic connection contains a pole with real part of residueless than or equal to −1 then the omega-limit set is exactlythe pole. Consequently, there is not a geodesic which is densein 𝑆.
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Strict analytic classification of simplest germs ofsemi-hyperbolic maps
Polina Shaikhullina
Chelyabinsk State University
A germ of a biholomorphic map 𝐹 : (C2, 0) → (C2, 0) is calledsemi-hyperbolic if one of its multipliers is equal to one andthe second one is hyperbolic. For example, the 1-time shift𝐹𝜆 = 𝑔1𝑣𝜆 along the vector field 𝑣𝜆 = 𝑥2 𝜕
𝜕𝑥 + 𝜆𝑦 𝜕𝜕𝑦 (𝜆 ∈ R+) is
semi-hyperbolic.
Two semi-hyperbolic germs will be called strictly equiva-lent if the local change of coordinates conjugating them hasthe form (𝑥 + 𝑜(𝑥2), 𝑦 + 𝑜(𝑥)), 𝑥 → 0. Let F𝜆 be the class ofsemi-hyperbolic germs strictly formally equivalent to the germ𝐹𝜆. We will show, that strict analytic classification of F𝜆 hasfunctional modules.
Consider the class 𝑀 consisting of collections(𝐴±, 𝐵±, 𝐴,𝐵,𝐶) such that: 𝐶 ∈ C; 𝐴+, 𝐵+ are holomorphicin (C,∞) × (C, 0) and 𝐴+(𝑡, 𝜏) = 𝑂(1), 𝐵+(𝑡, 𝜏) = 𝑂(𝑡−1),𝑡 → ∞; 𝐴, 𝐵 are holomorphic in (C, 0) and 𝐴(𝜏) = 𝑂(𝜏),𝐵(𝜏) = 𝑂(𝜏2), 𝜏 → 0; 𝐴−, 𝐵− are holomorphic in(C2, 0)𝐴−(𝑡, 𝜏) = 𝑂(𝑡2), 𝐵−(𝑡, 𝜏) = 𝑂(𝑡), 𝑡→ 0.
Theorem 1. The space 𝑀 is the space of functional modulesof strict analytic classification of germs of F𝜆.
Theorem 2. A germ 𝐹 ∈ F𝜆 has a holomorphic central man-ifold if and only if 𝐶 = 𝐵±(𝑡, 0) = 0 for corresponding compo-nents of its module.
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Denote by V𝜆 the class of strict formally equivalency ofthe 𝑣𝜆.
Theorem 3. A germ 𝐹 ∈ F𝜆 is embaddable into a flow (i.e., itis a 1-time shift along a field 𝑣 ∈ V𝜆) if and only if 𝐴± = 𝐵± =0 for corresponding components of its module. In this case, thecomponents 𝐶, 𝐴 are the Martinet-Ramis modules of the field𝑣[1], and 𝐵 is the Meshcheryakova-Teyssier module[2].
Theorem 4. Suppose that 𝐹 ∈ F𝜆 has a holomorphic centralmanifold (then restriction 𝑓 of the germ 𝐹 to the central man-ifold is a parabolic map). Then the non trivial restrictions ofthe components of the module of the germ 𝐹 to the line {𝜏 = 0}are the Ecalle-Voronin modules of the germ 𝑓 .
The work is supported by a grant RFBR 17-01-00739 A
REFERENCES
[1] Martinet J., Ramis J.-P. Probl‘eme de modules pour des ’equa-tions diff’erentielles non lin’eaires du premier ordre, Inst. Hautes’Etudes Sci.PublMath. (1982), 55.
[2] Voronin S. M., Meshcheryakova Yu. I. Analiticheskaya klassi-fikatsiya sedlouzlov, Tr. MMO (2005), 66.[Russian]
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Milnorian and non-Milnorian representations
Ilia Smilga
Higher School of Economics
In 1977, Milnor formulated the following conjecture: everydiscrete group of affine transformations acting properly on theaffine space is virtually solvable. We now know that this state-ment is false; the current goal is to gain a better understand-ing of the counterexamples to this conjecture. Every groupthat violates this conjecture “lives” in a certain algebraic affinegroup, which can be specified by giving a linear group and arepresentation thereof. Representations that give rise to coun-terexamples are said to be non-Milnorian. We will presentthe progress made so far towards classification of these non-Milnorian representations.
Various problems in parabolic equations withhysteresis
Sergey Tikhomirov
Saint-Petersburg State University
We study several problems with discontinuous hysteresis, of-ten called non-ideal relay. The main features of systems withhysteresis are the dependence of the output on the prehistoryof the input, rate independence, and nonsmoothness. Hys-teresis operators are applied in mathematical descriptions of
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various physical, chemical and biological processes: thermo-control, chemical reactors, ferro-magnetism, self-organisationand others.
We study periodic solutions in thermocontrol systems andpattern formation mechanisms for spatially distributed hys-teresis, which describes biological systems. We explore a newmechanism for pattern formation, which we called “rattling”.It seems the essence of the phenomena is related to the evolu-tion of the Young measures.
The Weotzenbock theorem about invariants
Henryk Zo ladek
University of Warsaw
We propose a new and constructive proof of the old theorem ofWeitzenbock that the ring of polynomial invariants of a linearvector field in C𝑚 is finitely generated.
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Finitely additive measures on the invariantfoliations of Anosov diffeomorphisms
Dmitry Zubov
National Research University Higher School of Economics,Moscow, Russian Federation
Let 𝐹 : 𝑀 → 𝑀 be a 𝐶2+𝑠 topologically mixing Anosov dif-feomorphism of a compact Riemannian manifold whose stableand unstable foliations are oriented and have dimensions 𝑑−𝑘and 𝑘. We construct a Banach space B of currents of degree 𝑘whose elements induce finitely additive measures on ’nice’ sub-sets of 𝑘-dimensional admissible leaves, whose tangent spaceeverywhere belongs to the unstable cone of the map 𝐹 . Allmeasures induced by the elements of B are regular in the fol-lowing sense: given a 𝐶2+𝑠 family 𝑋𝑡 of subsets of admissibleleaves with 𝐶1 boundary, the measure of 𝑋𝑡 is a 𝐶1 functionin 𝑡.
The constructed Banach space B contains the unique 𝜎-additive positive measure 𝜇𝑈 on the unstable foliation suchthat 𝐹 *𝜇𝑈 = 𝑒ℎ𝜇𝑈 , where ℎ is topological entropy of 𝐹 . Sinceany 𝐶1+𝑠 smooth function 𝑓 : 𝑀 → R is a bounded multiplierin B, we can consider the measure 𝑓 𝑑𝜇𝑈 as an element ofB. The integration of elements of B on nice subsets defines acontinuous linear functional on B, which allows to deduce thefollowing effective equidistribution result.
Theorem 1. If a 𝐶2+𝑠 topologically mixing Anosov diffeomor-phism 𝐹 : 𝑀 →𝑀 has oriented foliations and satisfies certainbounded distortion estimates, then there exists 𝜁 ∈ (0, 1) and a
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constant 𝐶 > 0 such that, for any 𝑓 ∈ 𝐶1+𝑠(𝑀) and any unitunstable ball 𝐵 = 𝐵(𝑥) centered at 𝑥 ∈𝑀 , we have
1
𝜇𝑈 (𝐹𝑛𝐵)
∫𝐹𝑛𝐵
𝑓 𝑑𝜇𝑈 −∫𝑀𝑓 𝑑m
≤ 𝐶 · 𝜁𝑛 · ‖𝑓‖𝐶1+𝑠(𝑀),
where m is the measure of maximal entropy of the map 𝐹normalized so that m(𝑀) = 1.
The key argument is the fact that the pullback operator𝐹 * is quasi-compact in B and has spectral radius 𝑒ℎ. Fur-thermore, we show that the Banach space B admits a finitedimensional subspace, spanned by the eigenvalues which arevery close to 𝑒ℎ, whose elements induce regular finitely addi-tive measures, which are invariant under the holonomy in thestable direction.
Acknowledgements. The work was supported by the Aca-demic Fund Program of the National Research UniversityHigher School of Economics (HSE) in 2018-2019 (grant number18-05-0019) and by the Russian Acamedic Excellence Project‘5-100’. The author is a Young Russian Mathematics awardwinner and would like to thank its sponsors and jury.
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