multiple returns for some regular and mixing maps...electronic mail: [email protected] electronic...
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Multiple returns for some regular and mixing maps
N. Haydn1 ∗, E. Lunedei2 †, L. Rossi2,3,4 ‡, G. Turchetti2,3 §, S. Vaienti4,5 ¶
(1) Department of Mathematics, University of Southern
California, Los Angeles 90089-1113, USA
(2) Dipartimento di Fisica, Universita di Bologna,
Bologna 40126, Italy
(3) INFN Sezione di Bologna, Bologna 40126, Italy
(4) UMR-CPT 6207, Universites d’Aix-Marseille I,II
et du Sud Toulon-Var, Marseille, France
(5) Federation de Recherche des Unites de Mathematique
de Marseille, Marseille, France
March 25, 2007
Abstract
We study the distributions of the number of visits for some noteworthy dynamical
systems, considering whether limit laws exist by taking domains that shrink around
points of the phase space. It is well known that for highly mixing systems such limit
distributions exhibit a Poissonian behaviour. We analyze instead a skew integrable map
∗Electronic mail: [email protected]†Electronic mail: [email protected]‡Electronic mail: [email protected]§Electronic mail: [email protected]¶Electronic mail: [email protected]
1
defined on a cylinder which models a shear flow. Since almost all fibers are given by
irrational rotations, we at first investigate the distributions of the number of visits for
irrational rotations on the circle. In this last case the numerical results strongly suggest
the existence of limit laws when the shrinking domain is chosen in a descending chain
of renormalization intervals. On the other hand, the numerical analysis performed for
the skew map shows that limit distributions exist even if we take domains shrinking in
an arbitrary way around a point, and these distributions appear to follow a power law
decay of which we propose a theoretical explanation. It is interesting to note that we
observe a similar behaviour for domains wholly contained in the integrable region of the
standard map. We consider also the case of two or more systems coupled together, proving
that the distributions of the number of visits for domains intersecting the boundary
between different regions are a linear superposition of the distributions characteristic of
each region. Using this result we show that the real limit distributions can be hidden
by some finite size effects. In particular, when a chaotic and a regular region are glued
together the limit distributions follow a Poisson-like law, but as long as the measure of
the shrinking domain is not zero the polynomial behaviour of the regular component
dominates for large times. Such analysis seems helpful to understand the dynamics in
the regions where ergodic and regular motions are intertwined, as it may happen for the
standard map. Finally, we study the distributions of the number of visits around generic
and periodic points of the dissipative Henon map. Although this map is not uniformly
hyperbolic, the distributions computed for generic points show a Poissonian behaviour, as
usually happens for systems with a highly mixing dynamics, whereas for periodic points
the distributions follow a different law which is obtained from the statistics of first return
times by assuming that subsequent returns are independent. These results are consistent
with a possible rapid decay of the correlations for the Henon map.
2
We extend to multiple returns the results, published in a previous issue of Chaos
[1], on the statistics of first return times for an integrable skew map defined on a
cylinder and for systems with different invariant components. For the skew map
we present a theoretical explanation of the observed power law decay of the dis-
tributions of the number of visits. In the case of systems which split into invariant
regions we prove that the distributions for domains intersecting the boundaries
are a linear superposition of the distributions characteristic of each region. These
results may allow to understand the dynamical behaviour of systems of physical
interest where regular and chaotic motions coexist, such as the standard map. We
show for this map, by an intense numerical analysis, that for any domain which
intersects the regular region and the chaotic sea the distributions of the number
of visits are well described by the superposition of a Poisson-like distribution and
of a power law. To conclude, we examine the dissipative Henon map. The dis-
tributions at generic points appear to follow a Poisson law, whereas at periodic
points they exhibit a different behaviour, which is obtained from the statistics of
first return times by assuming that subsequent returns are independent.
1 Introduction
The statistics of first return times has been extensively studied in the last years, mainly to
characterize the ergodic and statistical properties of dynamical systems. In two previous papers
[1, 2] we considered such a statistics for a skew map defined on a cylinder and describing a shear
flow. In this article we wish to extend our investigation to successive return times analyzing
the distributions of the number of visits for domains that shrink around a point of the phase
space.
Let T be a transformation on the space Ω and µ be a probability invariant measure on
Ω. Denoting by χA the characteristic function of a measurable set A ⊆ Ω, we can define the
3
“random variable” ξA in x (the symbol ⌊ . ⌋ represents the integer part)
ξA(t; x) =
⌊〈τA〉 t⌋∑
j=1
(χA T j
)(x). (1)
The value ξA(t; x) measures how many times a point x ∈ A returns into A after ⌊〈τA〉 t⌋
iterations of the map T . Here 〈τA〉 is the conditional expectation of the first return time τA(x)
of a point x, namely
τA(x) = min(
k ∈ N : T k(x) ∈ A∪ +∞
)(2)
and
〈τA〉 =
∫
AτA(x) dµA, (3)
where µA represents the conditional measure with respect to A of any measurable set B ⊆ Ω
µA(B) =µ(B ∩A)
µ(A). (4)
We recall that, for ergodic measures, 〈τA〉 is simply equal to µ(A)−1, as prescribed by Kac’s
theorem.
In what follows we will be interested in the distributions of the number of visits in A
Fk,A(t) = µA
(
x ∈ A : ξA(t; x) = k)
(5)
in the limit µ(A) → 0, denoting the limit distributions, whenever they exist, by
Fk(t) = limµ(A)→0
Fk,A(t). (6)
Of particular interest is the distribution of order k = 0: in this case Eq. (5) gives the statistics
(with respect to t) of first return times which we quoted at the beginning
F0,A(t) = µA
(
x ∈ A : τA(x)/〈τA〉 > t)
; (7)
we will set F (t) ≡ F0(t) the respective limit distribution, if it exists. Correspondingly, we
define the distribution of first return times as
Gr,A(t) = µA
(
x ∈ A : τA(x)/〈τA〉 ≤ t)
, (8)
4
calling Gr(t) the limit distribution, when it exists; of course Gr(t) = 1 − F (t).
The limit distributions Fk(t) have been analytically computed for dynamical systems with
strong mixing properties and whenever the set A is chosen as a ball or a cylinder (originating
from a dynamical partition of Ω) shrinking around µ-almost all points of Ω [3, 4, 5, 6, 7, 8, 9, 10];
in these highly mixing situations the asymptotic laws are Poissonian
Fk(t) =e−t tk
k!. (9)
The use of cylinders around arbitrary points requires different normalizations of the successive
return times; see for instance [4, 8] for a careful analysis of the statistics of first return times
around periodic points.
We will study the distributions of the number of visits for our skew map defined on the
cylinder C = T × [0, 1]
R :
x′ = x + y mod 1
y′ = y(10)
which is not ergodic with respect to the invariant Lebesgue measure. As we argued in [1], by
perturbing this simple map we obtain, according to the KAM theory, a transformation that
is integrable only for a subset of C whose Lebesgue measure approaches 1 as the amplitude of
the perturbation vanishes.
It is interesting to note that our skew map represents a shear of the linearized dynamics
at the fixed point of the following class of area preserving maps on T2 recently investigated in
[11]
x′ = x + h(x) + y
y′ = h(x) + ymod 1, (11)
where h is a smooth symmetric function vanishing at zero with its first derivative like, for
example, h(x) = x− sin(x). These maps are not uniformly hyperbolic and the authors showed
for them a polynomial decay of the correlations. Moreover, they argued that non-uniform
hyperbolicity produces “regions in which the motion is rather regular and where the systems
spend a substantial fraction of time (sticky regions)” [11]. This is also the physical framework
which motivated our previous work [1] and this one too, as we will point out again in Sec. 4.
Furthermore, the transformation (10) describes the flow on a square billiard [12].
5
Considering an arbitrary measurable domain A ⊆ C of positive Lebesgue measure, the
expectation (3) can be computed explicitly for the skew map
〈τA〉 =µy(Iy)
µ(A), (12)
where µx, µy represent the Lebesgue measure along the x-axis and y-axis, respectively, and
Iy = y : µx(Ay) > 0 (13)
with
Ay = (x′, y′) ∈ A : y′ = y . (14)
By choosing the set A as a square of side ǫ with a corner on the fixed point (0, 0), that is
A = [0, ǫ] × [0, ǫ], we proved in [1] that 〈τA〉 = 1/ǫ and that the limit law F (t) is
F (t) =
1 if t = 0
1
2if 0 < t ≤ 1
1
2t2if t > 1
. (15)
Of course, the same result holds for each of the fixed points (x, 0) since the map enjoys a
symmetry for translations. Moreover, through a numerical investigation we found that a similar
power law decay also holds for square domains centered around arbitrary points of the cylinder.
Before going to consider the distributions of the number of visits for the skew map R, we
observe that for almost all the ordinates y the dynamics along the fiber placed at y is given by
the irrational rotation x′ = x + y mod 1. Therefore the first natural step should be to inquire
whether limit distributions exist for irrational rotations. Since, as far as we know, surprisingly
there is not an answer to this question yet, in the next section we will investigate numerically
a special case, based on the analytical work of Coelho and De Faria [13] about the statistics
of first entry times. Then we will return to our skew map, giving a heuristic but quantitative
argument to claim the existence of limit distributions. Various applications will be presented
in Sec. 4, where in particular we will study the behaviour of the distributions of the number of
6
visits when two invariant regions are coupled together and for the standard map. The last part
of our work will be dedicated to show some results about the distributions for the dissipative
Henon map. Although this map is not area preserving like the others considered above, it
represents a very famous example of a two-dimensional chaotic system which is of interest
from a physical point of view too.
2 Irrational rotations
The existence of at most three different return times for irrational rotations on the circle was
proved by Slater about forty years ago [14]. However, the proof of the existence of a limit law,
provided the shrinking subsets of the circle are chosen in a descending chain of renormalization
intervals, is a recent result. Coelho and De Faria [13] were able to find the limit distribution
of first entry times when the rotation number is taken as an arbitrary irrational number (see
also [15] for further improvements), whereas a simple proof concerning the limit statistics of
first return times and restricted to a particular class of quadratic irrationals was independently
given by Turchetti [16].
2.1 Existence of limit laws
Let Rα(x) = x + α mod 1 be the irrational rotation with rotation number α. It is clear that
we may consider, without loss of generality, 0 < α < 1; so the continued fraction expansion
of the rotation number can be written like α = [0, a1, a2, a3, . . .]. The truncated expansion of
order n of α is given by pn/qn = [0, a1, . . . , an], where pn and qn verify the recurrence relations
pk = ak pk−1 + pk−2
qk = ak qk−1 + qk−2
(16)
with p−2 = 0, p−1 = 1 and q−2 = 1, q−1 = 0. Now, choosing an arbitrary point z on the circle,
we define An as the closed interval of endpoints Rqn−1
α (z) and Rqn
α (z) containing z; this means
that An = [Rqn
α (z), Rqn−1
α (z)] if n is odd, and the contrary holds if n is even. The sequence
of sets An represents the descending chain of renormalization intervals shrinking to z that are
used to prove the existence of a limit law.
7
Such a proof is especially simple when α is taken as a quadratic irrational with all the
coefficients of the continued fraction expansion equal, that is α = [0, a, a, a, . . .]. It is straight-
forward to show (see [16] the appendix) that when n is odd (even) the return time of a point
z ∈ An into the interval An is qn−1 (qn) if z < z, while it is qn (qn−1) if z > z. Therefore, using
the recurrence relations and taking into account that α(a + α) = 1, it is possible to prove that
the following limits exist
w1 = limn→∞
µ(A1n)
µ(An)=
α
1 + α, w2 = lim
n→∞
µ(A2n)
µ(An)=
1
1 + α(17)
t1 = limn→∞
qn−1
〈τAn〉
=α(1 + α)
1 + α2, t2 = lim
n→∞
qn
〈τAn〉
=1 + α
1 + α2(18)
where A1n and A2
n denote the subsets of An whose points return after qn−1 and qn iterations
respectively; note that w1 +w2 = 1 and t1 = α t2. The limit statistics of first return times then
reads
F (t) =
1 if 0 ≤ t < t1
w2 if t1 ≤ t < t2
0 if t ≥ t2
. (19)
As said before, the proof given by Coelho and De Faria concerns instead the limit distri-
bution of first entry times for arbitrary irrational rotation numbers. In this respect, the entry
time of a point x (belonging to the phase space Ω) into the set A ⊆ Ω is defined as its first
entrance τA(x) in A; the corresponding distribution is given by
Ge,A(t) = µ(
x ∈ Ω : τA(x)/〈τA〉 ≤ t)
. (20)
We will call Ge(t) the limit distribution for µ(A) → 0, when it exists. Let us now consider
in detail one of the two asymptotic laws found in [13]. If α = [0, a1, a2, a3, . . .], then it is well
known that an = ⌊1/Hn−1(α)⌋, where H : [0, 1[ → [0, 1[ is the Gauss transformation defined
as H(0) = 0, H(x) = 1/x for x > 0 (here . denotes the fractional part). By setting
bj = ⌊1/Hj−1(β)⌋, with β ∈ [0, 1[ and j ≥ 1, we can construct the following quantities
Γn(α, β) = (Hn(α), [0, an, an−1, . . . , a1, b1, b2, . . .]) , n ≥ 1. (21)
8
Of course the convergent subsequences of Γn(α, β) do not depend on β. Coelho and De Faria
proved in particular that for each subsequence σ = ni the distribution functions Ge,Ani(t)
converge to a continuous piecewise linear function Ge(t) if Γni(α, β)ni→∞−→ (θ, ω), with θ ∈ ]0, 1]
and ω ∈ [0, 1[. Note that the intervals An are defined in the same way as seen above. Since
we are mostly interested in return times, we would like to obtain, starting from the knowledge
of Ge(t), the limit statistics of first return times F (t), which represents the zero order (k = 0)
distribution of the number of visits. This can be done by using a very recent result by Lacroix
et al. [17] that establishes the following relation between Ge(t) and F (t)
Ge(t) =
∫ t
0
F (s) ds. (22)
Thus, considering the expression of Ge(t) found by Coelho and De Faria, we have
F (t) =
1 if 0 ≤ t < ta
1
1 + ωif ta ≤ t < tb
0 if t ≥ tb
, (23)
with
ta =ω(1 + θ)
1 + θω, tb =
(1 + θ)
1 + θω. (24)
This result allows us to hope that also the limit distributions of higher order exist. Note that
the statistics (23) is consistently reduced to the one given by Eq. (19) if α = [0, a, a, a, . . .],
since in this case θ = α and ω = α.
2.2 Numerical analyses
In order to numerically investigate the limit distributions of the number of visits, we took the
rotation number equal to the golden ratio γ =√
5−12
, which exhibits the very simple continued
fraction expansion γ = [0, 1, 1, 1, . . .]. It is easy to see that, in this case, Γn(γ, ·) converges to
(θ, ω) = (γ, γ). The analysis has been performed for several orders k, computing for each of
them the distribution Fk,An(t), with n form 10 to 20. Of course it is not possible to deal with
9
0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5
Fk,
A20
(t)
t
k = 1k = 2k = 3
Figure 1: Distributions of the number of visits
Fk,A20(t) of order k = 1, 2, 3.
0
0.2
0.4
0.6
0.8
1.0
1.2
2 3 4 5 6 7
Fk,
A20
(t)
t
k = 3k = 4k = 5
Figure 2: Distributions of the number of visits
Fk,A20(t) of order k = 3, 4, 5.
limit distributions by means of numerical methods; nevertheless the results obtained strongly
suggest the existence of the limit distributions Fk(t). We found in fact that the distributions
Fk,An(t) with the same order k are very close to each other regardless of the value of n, and this
despite the presence of statistical fluctuations and effects due to the finite size of the intervals
An (the measure of An goes from about 2× 10−2 for n = 10 to 2× 10−4 for n = 20). Since we
are interested in the limit for µ(A) → 0, in Fig. 1, 2 are shown the distributions referring to
the smaller interval only, namely A20; however, the distributions of the same order computed
for different values of n would appear practically indistinguishable in the graph.
It is worthwhile to note some of the features of the distributions Fk,An(t) numerically
obtained. First, their support is an interval and, for any k, it can be partitioned in three
subintervals I(l)k , I
(c)k and I
(r)k (the leftmost, the central and the rightmost respectively) in
such a way that Fk,An(t) is constant on each of these subintervals; in particular Fk,An
(t) = 1
if t ∈ I(c)k . Furthermore, the intervals I
(r)k and I
(l)k+1 practically coincide (in this regard, the
distribution with k = 3 is reported in both figures to clearly show that this is true even for
I(r)2 , I
(l)3 and I
(r)3 , I
(l)4 ). For every distribution studied we have that µ(I
(l)k ) ≃ µ(I
(r)k ) ≃ 0.447
and µ(I(c)k ) ≃ 0.724, except for k = 2 and k = 5 where µ(I
(c)k ) ≃ 0.276. Interestingly enough,
the measure of the support of Fk,An(t) for k = 1, 3, 4 is about 1.618, that is near to 1/γ.
As an example, we report the distributions Fk,A20(t) of order one and two (in the interval
10
in which they differ from zero)
F1,A20(t) =
0.38 if 0.72 ≤ t < 1.17
1 if 1.17 ≤ t < 1.89
0.24 if 1.89 ≤ t < 2.34
(25)
and
F2,A20(t) =
0.76 if 1.89 ≤ t < 2.34
1 if 2.34 ≤ t < 2.62
0.85 if 2.62 ≤ t < 3.06
. (26)
It is our opinion that it would be interesting to get an analytic proof about the existence
of the limit distributions Fk(t) and an explanation of their properties.
3 Shear flow
The statistics (23) found for the irrational rotations on the circle is one of the two that can
be obtained, for each k, by looking at the renormalization intervals An. On the other hand,
nothing is known if one considers, for example, intervals of length ǫ going continuously to zero.
What is surprising is that for our skew map (10), which is horizontally almost everywhere
foliated by irrational rotations, a limit distribution exists for the first return time when we
shrink the square domain Aǫ = [0, ǫ] × [0, ǫ] toward the fixed point (0, 0); thus one could
wonder if, in such a situation, the limit distributions of the number of visits exist for a generic
order k > 0. In this case however a geometric proof, as was performed for the first return times,
is exceedingly complicated; nevertheless the numerical computations suggest that a limit law
still exists, as we will see in a moment. To understand this fact, we present here a heuristic, but
quantitative, argument which provides predictions very close to the numerical observations.
In this respect, we have to consider another equivalent characterization of the distributions
11
of the number of visits. Let us begin to introduce the k-th return time in A of a point x ∈ A
τkA(x) =
0 if k = 0
τk−1A (x) + τA
(
T τk−1
A(x)(x)
)
if k ≥ 1
(27)
(note that τ 1A(x) = τA(x)) and define the distribution of the k-th return time as
Pk,A(t) = µA
(
x ∈ A :τkA(x)
〈τA〉≤ t
)
. (28)
We then observe that Eq. (5) can be rewritten as [6]
Fk,A(t) = µA
(
x ∈ A :τkA(x)
〈τA〉≤ t ∧
τk+1A (x)
〈τA〉> t
)
= Pk,A(t) − Pk+1,A(t). (29)
Since
τkA = τA + (τ 2
A − τA) + . . . + (τkA − τk−1
A ), (30)
the function Pk,A(t) represents also the distribution of the sum, normalized by 〈τA〉−1, of the
differences of consecutive return times until the k-th return. We remark that the distribution
of the difference between two consecutive return times (normalized by 〈τA〉−1) follows the same
law as the distribution of the first return [6], because the measure µA is invariant with respect
to the induced application on A and
τkA − τk−1
A = τA T τk−1
A . (31)
If the random variables τA/〈τA〉, (τ 2A−τA)/〈τA〉, . . . , (τk
A−τk−1A )/〈τA〉 were i.i.d. with the same
distribution function Gr,A(t), it is well known that the distribution function of their sum would
be the following convolution product
Pk,A(t) = Gr,A(t) ∗ Gr,A(t) ∗ . . . ∗ Gr,A(t)︸ ︷︷ ︸
k times
. (32)
In the case of highly mixing systems (for instance φ, α and (φ, f) mixing systems, see [6, 8, 10])
for which the limit distribution of first return times Gr(t) is almost everywhere given by 1−e−t,
12
the differences of the normalized successive return times become asymptotically independent
when µ(A) → 0; the strategy adopted in [6] to obtain the Poisson law
Pk,A(t) − Pk+1,A(t) −→e−t tk
k!, (33)
(for a suitable choice of the sets A, as said in Sec. 1) was just based on this fact.
Now we proved in [1] that, although our skew map is not ergodic, it enjoys a “local”
mixing-like property: for square domains Aǫ = [0, ǫ] × [0, ǫ] we showed that
∣∣µǫ(Aǫ ∩ T n(Aǫ)) − µ2
ǫ (Aǫ)∣∣ = O(n−1) , (34)
where µǫ = µ/ǫ. This suggests to try to get the distributions of the number of visits by
assuming that the differences of successive return times are asymptotically independent. We
know that the limit statistics of first return times F (t), obtained for the sets Aǫ when ǫ → 0,
is given by Eq. (15). Being Gr(t) = 1 − F (t) the corresponding limit distribution, under the
preceding assumption we can write
Fk(t) = Pk(t) − Pk+1(t) (35)
with
Pk(t) = Gr(t) ∗ Gr(t) ∗ . . . ∗ Gr(t)︸ ︷︷ ︸
k times
. (36)
In particular, we have that
F1(t) = Gr(t) −
∫ +∞
−∞Gr(t − s) dGr(s), (37)
and a rather straightforward computation of the Stieltjes integral gives
F1(t) =
0 if t = 0
1/4 if 0 < t < 2
1
4t2+
1
4(t − 1)2+
3
2t3+
3 log(t − 1)
t4+
6 − 7t
4t3(t − 1)2if t ≥ 2
. (38)
13
Note that when t is large, F1(t) behaves like 1/2 t−2. With a similar but more cumbersome
computation we can obtain F2(t)
F2(t) =
0 if t = 0
1/8 if 0 < t < 1
1
4−
1
8t2if 1 ≤ t < 2
O(t−2
)if t ≫ 2
(39)
Using a recursive argument it is possible to show two interesting features of the distributions
Fk(t):
– for 0 < t < 1 the distributions present a plateau whose height is given by 1/2k+1, and
this is the only explicit dependence on k that we are able to easily detect;
– for t → ∞ all the Fk(t) exhibit the same behaviour, decaying like 1/2 t−2.
We compared these distributions, computed under the assumption that the differences of suc-
cessive return times are asymptotically independent, with the ones obtained through the nu-
merical analysis. Although there is some discrepancy between them, nonetheless the qualitative
features described above still seem to persist. In particular, both types of distribution show an
initial plateau, even if the actual one decreases to zero much faster for k > 1 and, what is more
interesting, all the numerical distributions appear to decay like 1/2 t−2 after a transitory peak,
see Fig. 3, 4. The discrepancy between the two kind of distribution could be reasonably due to
the presence of some sort of weak correlation between the differences of successive returns. We
met an analogous situation in the previous section, where the same asymptotic independence
assumption was made for the irrational rotations on the circle: again we did not recover the
numerical distributions for k = 1, 2, although we were able to depict the qualitative behaviour
of the limit laws. Note that this could be in agreement with a result of Coelho and De Faria
[13], which shows that for θ > 0 the limit joint distributions of the differences of successive
entry times are not given by the product of the individual limit distributions.
14
10-8
10-6
10-4
10-2
100
10-1 100 101 102 103 104
F1(
t)
t
Figure 3: Distribution of the number of visits of
order k = 1 computed for the domain Aǫ = [0, ǫ]×
[0, ǫ] with ǫ = 10−2. The dotted line represents the
function (38).
10-8
10-6
10-4
10-2
100
10-1 100 101 102 103 104
F2(
t)
t
Figure 4: Distribution of the number of visits of
order k = 2 computed for the domain Aǫ = [0, ǫ]×
[0, ǫ] with ǫ = 10−2. The dashed line represents
the function 1/2 t−2.
We also investigated the behaviour of the distributions of the number of visits for sets A =
[0, ǫ] × [y0, y0 + ǫ] with y0 > 0, using a least-squares method to fit the numerical distributions,
obtained for several values of y0 and ǫ, against the function α t−β . We found that an asymptotic
power law decay preceded by a peak seems to hold even for such domains, as shown in Fig. 5;
note how the peaks narrow and shift toward larger t when k increases, while their height
slightly decreases. In this more general case however the exponent β is usually greater than
2 and the mean value of its distribution appears to be positively correlated to the order k
(Fig. 6). Even for rectangular domains like A = [x0, x0 + ǫ] × [y0, y0 + δ], the distributions
numerically computed present similar features: in particular their decay follows a power law
with an exponent greater but near to 2 and the mean return time still is 〈τA〉 = 1/ǫ
In conclusion, we think that the inverse square decay in t, whatever the order k, is typical for
the fixed points (which lie along the x-axis) of our skew map, while as soon as we consider other
points the exponent β changes weakly with k. We wish to remark that the differences between
the distributions for periodic and generic points seem to be a general fact of recurrences, as we
will also argue in the next sections.
15
10-6
10-4
10-2
100
100 101 102 103
Fk(
t)
t
1 2 5 10 20
Figure 5: Distributions of the number of visits of
order k = 1, 2, 5, 10, 20 computed for the domain
A = [0, ǫ]×[y0, y0+ǫ] with y0 = 0.35 and ǫ = 10−2.
The dashed line represents the function 1/2 t−2.
1.95
2.00
2.05
2.10
2.15
k = 1 k = 2 k = 3
β
Figure 6: Distributions of the best-fit parameter
β obtained through a least-squares method, in the
range 30 ≤ t ≤ 100, from the distributions of the
number of visits of order k = 1, 2, 3. For each k,
the distributions of the number of visits have been
computed for twenty domains of side ǫ = 10−2.
The arrows show the position of the mean value of
β.
4 Mixed dynamical systems
We studied the skew map (10) both in [1] and in this paper mainly for two reasons. First, we
tried to understand, by means of a very simple model, why the statistics of first return times
exhibits a power law decay for systems which present regular components, as it was pointed
out, for example, in [18, 19, 20, 21] (see also [22, 23] for an overview of recurrences in dynamical
systems).
The second reason was to investigate whether this polynomial decay could be related to
some finite size effect. In this respect, we considered in [1] a system whose phase space was
partitioned into two invariant regions, obtained by coupling the skew map, which models a shear
flow, with some mixing map whose statistics of first returns decays exponentially. We were able
to rigorously prove the following fact: for a particular domain A that intersects the boundary
16
of the two invariant regions and whose measure goes to zero, the limit statistics of first return
times is simply ruled by the one of the mixing region, thus being exponential. Nevertheless,
as long as the measure of A has a finite positive value (which is the only situation that can
be analyzed through numerical investigations), the exponential and power laws are linearly
superposed and weighted by coefficients equal to the relative sizes of the chaotic and regular
components of A; in this case the power law asymptotically provides the main contribution.
The proof relies on the fact that, for the particular domain considered, we know analytically
the statistics of first return times, and this allow us to understand how the statistics converges
to its limit behaviour. Unfortunately we do not have such a knowledge for the distributions of
order k > 0.
Here we will present a general result about the linear weighted superposition of the distri-
butions of the number of visits when two invariant regions are coupled, which generalizes what
we previously proved for the statistics of the first return [1]. Such a result will be numerically
checked for a one-dimensional system composed by two mixing maps. Similar investigations
will also be performed for a two-dimensional system obtained by coupling our skew map with
the Arnold’s cat map and then for the so called ‘standard map’.
4.1 Distributions of the number of visits
Let T be a map acting on the measurable space Ω and µ be a T -invariant measure. Sup-
pose moreover that the dynamical system (Ω, T, µ) splits into two subsystems (Ω1, T1, µ) and
(Ω2, T2, µ), where
Ω = Ω1 ∪ Ω2, µ(Ω1 ∩ Ω2) = 0 (40)
and the maps T1, T2, defined over Ω1, Ω2 respectively, satisfy the following conditions
T1 = T|Ω1\(Ω1∩Ω2), T2 = T|Ω2\(Ω1∩Ω2) (41)
that is they coincide with T except possibly on the zero measure boundary Ω1 ∩ Ω2.
We wish to consider the behaviour of the distributions of the number of visits for neigh-
borhoods of points belonging to the common boundary of the invariant regions. To this end,
17
we take a neighborhood A of a point x ∈ Ω1 ∩ Ω2 such that µ(A) > 0, and denote with A1
and A2 the two different components of A, that is A = A1 ∪ A2, where A1 = A ∩ Ω1 and
A2 = A ∩ Ω2. We will choose the sequence of sets A shrinking around x in such a way that
the relative weights
w1(A) =µ(A1)
µ(A), w2(A) =
µ(A2)
µ(A)(42)
have a finite limit when µ(A) → 0, namely we will assume that the following limits exist and
are different from zero
w1 = limµ(A)→0
w1(A), w2 = limµ(A)→0
w2(A). (43)
By a procedure like the one we used in [1] it is easy to prove that the mean return time in A
is related to those in A1 and A2 as follows
〈τA〉 = w1(A) 〈τA1〉 + w2(A) 〈τA2
〉. (44)
Let now Fk,A1(t) and Fk,A2
(t) be the distributions of the number of visits in A1 and A2
respectively
Fk,Ai(t) = µAi
(
x ∈ Ai : ξAi(t; x) = k
)
, i = 1, 2; (45)
it is straightforward to show that
Fk,A(t) = w1(A) Fk,A1(w′
1(A) t) + w2(A) Fk,A2(w′
2(A) t) (46)
where
w′1(A) =
〈τA〉
〈τA1〉, w′
2(A) =〈τA〉
〈τA2〉. (47)
Assuming that the limits
Fk,1(t) = limµ(A)→0
Fk,A1(t), Fk,2(t) = lim
µ(A)→0Fk,A2
(t) (48)
and
w′1 = lim
µ(A)→0w′
1(A), w′2 = lim
µ(A)→0w′
2(A) (49)
are well defined, it is possible to prove, in a similar way as we did for the statistics of first
return times, that
18
Proposition 4.1 Under the existence of (43), (48) and (49), the limit distributions of the
number of visits exist and are given by
Fk(t) = w1 Fk,1(w′1 t) + w2 Fk,2(w
′2 t) (50)
at the points of continuity of both Fk,1 and Fk,2.
4.2 Coupling of chaotic maps
In order to check Eq. (50), we consider the following one-dimensional map defined over the
interval Ω = [−1, 1]
T (x) =
T1(x) if −1 ≤ x < 0
T2(x) if 0 ≤ x ≤ 1
, (51)
where
T1(x) =
−3x − 3 if −1 ≤ x < −23
3x + 1 if −23≤ x < −1
3
−3x − 1 if −13≤ x ≤ 0
(52)
and
T2(x) =
3x if 0 ≤ x < 13
2 − 3x if 13≤ x < 2
3.
3x − 2 if 23≤ x ≤ 1
(53)
Note that the two subsets Ω1 = [−1, 0] and Ω2 = [0, 1] are invariant as regards T1 and T2
respectively.
The map T preserves the Lebesgue measure and is discontinuous at the point x = 0, which
is a periodic point of period two for T1 and a fixed point for T2. Thus, in order to deal with
Proposition 4.1, we need to know the distributions of the number of visits around periodic
points. Since the two mixing maps T1 and T2 are conjugated with a Bernoulli shift on three
19
symbols with equal weights, we can use a general formula recently proved by Haydn and Vaienti
which gives the limit distribution of order k for cylinders Cn around periodic points of period
P
Fk(t) = limn→∞
µCn
(
x ∈ Cn : ξCn(t; x) = k
)
= (1 − pP ) e−(1−pP )tk∑
j=0
(k
j
)pP (k−j) (1 − pP )2j
j!tj , (54)
where p = µ(Cn+1)/µ(Cn). Note that for k = 0 this formula coincides with the one found by
Hirata [4]
F0(t) = (1 − pP ) e−(1−pP ) t. (55)
If we consider, for example, a sequence of shrinking cylinders Cn centered around the point
x = 0 (so w1 = w2 = 1/2 and, by Kac’s theorem, w′1 = w′
2 = 1), the distribution of the number
of visits for k = 1 should be well described by the following function
F1(t)∣∣∣x=0
=1
2e−(1−p2)t (1 − p2)
[p2 + (1 − p2)2 t
]+
+1
2e−(1−p)t (1 − p)
[p + (1 − p)2 t
], (56)
where p = 1/3 in the present case. As Fig. 7 shows, the agreement of the numerically computed
distributions with the theoretical expectations is really good.
As a final interesting remark, we would stress the fact that the distributions (54) can be ob-
tained, under the assumption that the differences of successive return times are asymptotically
independent, by convoluting F0(t) according to the procedure described in Sec. 3.
4.3 Coupling of regular and mixing maps
We wish now to couple the skew map (10) with the hyperbolic automorphism of the torus
(Arnold’s cat map) by constructing a transformation T on the two-dimensional space Ω =
Ω1 ∪Ω2, where Ω1 = T × [0, 1] and Ω2 = T × [−1, 0[ , so that Ω is obtained by gluing together
the cylinder Ω1 with the torus Ω2. The map T1 = T|Ω1is represented by our skew map, while
T2 = T|Ω2is the hyperbolic automorphism
T2 :
x′ = 2x + y mod 1
y′ = (x + y mod 1) − 1 .(57)
20
100
10-1
10-2
10-3
10-4
10-5
10-6
0 5 10 15 20 25
Fk(
t)
t
1
2
Figure 7: Distributions of the number of visits of order k = 1, 2 computed for the map given
by Eq. (51) in the interval A = [−ǫ, ǫ] with ǫ = 5 × 10−4. The dotted lines represent the
theoretical predictions; in particular the one for k = 1 corresponds to formula (56).
Let us consider an arbitrary point Q = (x, 0) along the common boundary, taking as
neighborhoods around Q the square domains A = [x − ǫ/2, x + ǫ/2] × [(λ − 1)ǫ, λǫ] of side
ǫ, with 0 < λ < 1 fixed. Following the same notations introduced in Sec. 4.1, we have
A1 = [x− ǫ/2, x + ǫ/2]× [0, λǫ] and A2 = [x− ǫ/2, x + ǫ/2]× [(λ− 1)ǫ, 0[ ; thus using Eq. (42)
and Eq. (47) it is straightforward to show that
w1(A) = λ, w2(A) = 1 − λ (58)
and
w′1(A) = λ +
1
ǫ, w′
2(A) = (1 − λ)(1 + λǫ) . (59)
Although we do not exactly know the distributions of the number of visits Fk,A1(t) for A1,
nevertheless the numerical computations of Sec. 3, supported by the heuristic explanations
presented there, strongly suggest that Fk,A1(t) will decay, at least for A sufficiently small,
like α t−β, with β slightly greater than 2. Similar considerations can be done about Fk,A2(t):
since the cat map, which is an Anosov diffeomorphism, enjoys the Poisson distribution (if
we take balls or cylinders converging around almost all points), it is sensible to expect that
the distributions Fk,A2(t) will approach the function e−t tk/k! when µ(A) ≪ 1. Thus, using
21
10-8
10-6
10-4
10-2
100
100 101 102 103
F1(
t)
t
a
b
Figure 8: Distributions of the number of visits of order k = 1 computed for a square domain
of side (a) ǫ = 0.2 and (b) ǫ = 0.05, such that λ = 0.5. The dotted lines represent Eq. (60)
with α = 0.5 and β = 1.94. Note that as the domain’s size decreases, the contribution of the
regular component begins to prevail for larger values of t.
Eq. (46), we can write for t sufficiently large
Fk,A(t) = w1(A)α
(w′1(A) t)β
+ w2(A)(w′
2(A) t)k
k!e−w′
2(A) t (β ≃ 2). (60)
This formula, despite the approximations employed in order to obtain it, describes quite well
the asymptotic behaviour of the distributions of the number of visits for domains that intersect
the boundary, as shown in Fig. 8. Note that a power law tail appears as soon as the first term
in Eq. (60) gives the main contribution. Furthermore, the values of the normalized time t
for which the polynomial decay prevails increase as the measure of A decreases (but is still
different from zero); this means that if we numerically compute the distributions of the number
of visits for a very small domain, we need to reach large values of t to see the power law tail.
We wish to stress that the appearance of this power law tail is a consequence of the fact that
µ(A) 6= 0. Therefore, since it is not possible to deal with the limit of zero measure domains
while performing a numerical analysis of the distributions, one should take into account the
presence of these finite size effects before going to conclusions about the behaviour of the limit
distributions from the numerical results, as we pointed out in our previous work [1].
Now it would seem that we could directly turn to Proposition 4.1 to get the limit dis-
22
tributions Fk(t), but unfortunately it happens that one of the assumptions required for the
existence of Fk(t) fails, since w′1(A) has not a finite limit when µ(A) = ǫ2 → 0. We were
able to overcome this difficulty in the particular case presented in [1] because we knew how
the statistics of first return times for the skew map approaches its limit law: this allowed us
to directly prove that the behaviour of the mixing region prevails when µ(A) → 0. Here we
observe that, by supposing the quantity α keeps a finite positive value, the limit of Eq. (60)
for µ(A) → 0 simply reads
Fk(t) = w2(w2 t)k
k!e−w2 t , (61)
since limµ(A)→0 w′2(A) = w2. As we can see, even if the power law contribution disappears from
the distributions in the limit of zero measure domains, nevertheless the presence of the regular
component of A is still revealed by the fact that the coefficient w2 is different from one.
4.4 Standard map
In this section we investigate the distributions of the number of visits for the standard map
y′ = y −η2π sin(2πx)
x′ = x + y′
mod 1, (62)
choosing the coupling parameter as η = 3, such that the boundary between the integrable
orbits and the chaotic region is as sharp as possible compared to the size of the sets used to
compute the distributions. We consider three cases: domains wholly contained in the regular
region, domains lying on the chaotic ‘sea’ and domains which overlap both regions.
4.4.1 Regular region
The numerical distributions obtained for domains wholly contained in the regular region seem
to follow asymptotically a power law decay like α t−β, as shown in Fig. 9. The least-squares
fit estimate of the exponent β gives a value near 2, usually between 1.90 and 2.05, and there
is some evidence suggesting that β slightly increases along with the order k.
23
10-6
10-4
10-2
100
100 101 102 103
Fk(
t)
t
Figure 9: Distributions of the number of visits
of order k = 1 (lower curve), k = 2 (middle
curve) and k = 3 (higher curve), computed for
a square domain of side ǫ = 2.5 × 10−2 centered
at (0.15, 0.25), which is wholly contained in the
regular region. The distributions appear to follow
asymptotically a power law decay like α t−β , where
the least-square fit estimate of β, performed for t
in the range [10, 500], is 1.91 for k = 1, 1.95 for
k = 2 and 1.97 for k = 3.
10-8
10-6
10-4
10-2
100
100 101 102 103
Fk(
t)
t
12
Figure 10: Distributions of the number of visits of
order k = 1, 2 computed for a square domain of
side ǫ = 5 × 10−2 centered at (0.7, 0.2), which lies
on the chaotic sea. The dotted line represents the
function α t−β , with α = 10−2 and β = 2.
4.4.2 Chaotic region
The behaviour of the distributions of the number of visits for domains lying on the chaotic
component of the phase space appears at first surprising. In fact, the computed mean return
time is given, with a very good approximation, by the ratio of the measure of the chaotic region
(which is about 0.88 with the choice η = 3 for the coupling parameter) and the measure of the
considered domain, thus being in agreement with the value that would be provided by Kac’s
theorem (which holds for ergodic systems) if it could be applicable here. On the other hand,
the distributions obtained reveal a departure from the expected Poisson law (9) for sufficiently
large values of t, showing a polynomial decay with an exponent near 2 (see Fig. 10).
The reasons of such a rather unexpected feature could sensibly be due to the fact that
24
orbits originating even from points far away from the regular region usually approach it. In
this way, the overall chaotic motion would be appreciably influenced by the regular component,
leading to the appearance, in the distributions of the number of visits, of the observed power
law decay which is typical of integrable systems. In this regard it is interesting to note that
Poincare recurrences seem to provide a tool capable to capture some of the properties of the
dynamics in a more “sensitive” way with respect to other quantities that could be used for the
same purpose, like the mean return time into a given domain seen above.
4.4.3 Mixed regions
Considering the results of Sec. 4.1, it is reasonable to expect that the asymptotic behaviour
of the distributions of the number of visits for domains that intersect the boundary between
the regular and the chaotic regions should be given by a linear superposition of a power law
(the contribution, above all, of the integrable orbits) and a of Poisson distribution (from the
chaotic sea).
In order to test this conjecture, we tried to fit the distributions computed for a given domain
A against the following function, which should sensibly represent their behaviour (from now
on, to simplify the notations, we drop the explicit dependence on A)
Fk(t) = w1α
(w′1t)
β+ w2
(w′2t)
k e−w′2t
k!. (63)
From Eq. (42), (44) and (47) we have that the quantities w1, w2, w′1 and w′
2 are related in the
following way
w1 + w2 = 1,w1
w′1
+w2
w′2
= 1; (64)
so, for instance, w2 and w′2 in Eq. (63) can be expressed in terms of w1 and w′
1
w2 = 1 − w1, w′2 =
1 − w1
1 − w1/w′1
. (65)
Using then w1, w′1, α and β as fit parameters, we generally found a very good agreement
between the numerical distributions and formula (63), despite the fact that it does not take
into account the domain’s finite size effects. In this respect, Fig. 11 shows a distribution of
order k = 3 along with the corresponding fit function.
25
10-6
10-4
10-2
100
100 101 102 103
F3(
t)
t
Figure 11: Distribution of the number of visits of order k = 3 for a square domain of side
ǫ = 0.1 centered at (0.03, 0.3), which intersects the boundary between the regular and the
chaotic regions. The dotted line represents Eq. (63) with β = 2.04.
It is worthwhile to mention that, for a given domain A, generally the value w1 obtained
from the fit procedure is greater than the geometrical estimate of the relative measure of the
regular share of A. This discrepancy seems reasonably due to the fact that besides the points
belonging to the regular share, which are responsible for the power law decay, there is a further
contribution from the chaotic component of A, whose dynamics is influenced in some way by
the integrable orbits, as seen in Sec. 4.4.2, so that the effective relative size of the regular
component appears to be greater than expected.
5 Dissipative Henon map
In this section we will consider the following map defined on R2, which was introduced by
Henon [24]
x′ = 1 + y − ax2
y′ = bx
, (66)
where a = 1.4 and b = 0.3. In order to investigate the distributions of the number of visits
Fk(t), we used the so called physical measure (also known as SBR-measure) supposing it is well
defined, despite the fact that no rigorous proof about its existence and its statistical properties
26
10-6
10-4
10-2
100
0 5 10 15
Fk(
t)
t
0
1
2
Figure 12: Distributions of the number of visits of
order k = 0, 1, 2 computed for a circular domain
of radius 5 × 10−3 around a generic point. The
dotted lines represent the corresponding Poisson
laws.
10-5
10-4
10-3
10-2
10-1
100
0 10 20 30 40 50 60
Fk(
t)
t
515 30
Figure 13: Distributions of the number of vis-
its of order k = 5, 15, 30 computed for a circu-
lar domain of radius 5 × 10−3 around a generic
point. The dotted lines represent the correspond-
ing Poisson laws.
has been given until now. The numerical analyses were performed by taking the domain A as
a little ball centered both around generic and periodic points.
In the case of generic points, the distributions appear to agree in a very good way with the
Poisson law (9), as Fig. 12, 13 show. Since the Henon map is not uniformly hyperbolic, we
could not have taken for granted this result.
To study the distributions of the number of visits for periodic points we computed, by a
bisection method [25], the periodic orbits of the Henon map with period P from one to ten.
We limited our investigation up to points of period ten because the lower periods are the ones
with a stronger influence on return times. As expected, when A is a little ball centered around
a periodic point, the Poisson law (9) does not fit anymore the numerical results, as it is clear
from Fig. 14, which shows the distribution F0(t) obtained for a point of period two. Instead,
for every periodic point considered, the distributions of order k = 0 are well described by the
function (compare it to Eq. (55))
F0(t) = α e−αt, (67)
27
where the coefficient α seems to depend on the period P in the following way
α(P ) = 1 − AP (68)
with, according to our latest estimate, A = 0.96 ± 0.02 and = 0.59 ± 0.01 (see Fig. 16).
Then, by making the assumption of independence of the differences of successive return times,
we used a procedure similar to the one employed in Sec. 3 to obtain the distributions of the
number of visits of any order
Fk(t) = α e−αt
k∑
j=0
(k
j
)(1 − α)k−j α2j
j!tj . (69)
As Fig. 15 shows, the numerical distributions computed for periodic points agree very well with
this expression, which, under the identification α → (1− pP ), exactly coincides with Eq. (54);
as in that case, it would be interesting here to explicitly relate α to the period. Moreover,
we can note that Eq. (69) is reduced to formula (9) when α → 1. Since α approaches 1
as the period increases, this means that the distributions for high periods are practically
indistinguishable from those concerning generic points, which are just described by Eq. (9).
This is self-consistent, because an aperiodic orbit can be thought like a periodic orbit with an
infinite period.
Summarizing our work on the Henon map, we observe the following facts. First, the distri-
butions of the number of visits computed for generic points agree with the Poisson law (9), as
usually happens for systems with strong mixing properties (see [6]). Second, periodic points
exercise a real effect on return times, and this effect decreases as the period grows. Further-
more, the distributions computed for periodic points follow in a very good way the behaviour
predicted by assuming that the differences of successive return times are independent.
6 Conclusions
The distributions of the number of visits, whose definition is based on successive return times,
represent an extension of the statistics of first return times.
In order to investigate their behaviour for systems characterized by a regular dynamics,
we have studied at first a skew integrable map defined on a cylinder which is area-preserving
28
10-8
10-6
10-4
10-2
100
0 2 4 6 8 10 12 14 16
F0(
t)
t
a
b
Figure 14: Distribution of the number of visits of
order k = 0 computed for a circular domain of
radius 5 × 10−3 centered around a periodic point
of period two. The dotted line (a) corresponds
to the exponential function e−t, while the line (b)
represents the function α e−αt with α ≃ 0.66.
10-5
10-4
10-3
10-2
10-1
100
0 10 20 30 40 50 60 70
Fk(
t)
t
15
15 30
Figure 15: Distributions of the number of visits of
order k = 1, 5, 15, 30 computed for a circular do-
main of radius 5×10−3 centered around a periodic
point of period two. The dotted lines represent
the corresponding theoretical predictions (69).
and models a shear flow. Despite this map is horizontally almost everywhere foliated by
irrational rotations (which seem to admit piecewise constant limit distributions only if the
shrinking domain is chosen in a descending chain of renormalization intervals), the analysis of
the distributions of the number of visits Fk,A(t) suggests the existence of the corresponding limit
laws Fk(t) when A shrinks in an arbitrary way around points of the cylinder. In particular, for
square sets containing the fixed points of the map, the distributions present an initial plateau,
whose height decreases as the order k increases, and asymptotically decay like 1/2 t−2. Since
the same features can be found in the distributions computed by assuming that the differences
between successive return times are independent, we believe that, although our skew map is
not ergodic, the “local” mixing-like property it enjoys could play here some role. Moreover, an
asymptotic power law decay α t−β also holds for arbitrary rectangular domains not containing
the fixed points, even if in this case the exponent β is usually greater than 2 and seems to grow
along with the order k.
For systems composed by invariant regions we have extended the results previously ob-
29
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0 2 4 6 8 10 12
α(P
)
P
Figure 16: Values of the coefficient α of the distributions F0(t) computed for periodic points
of period P . The dotted line represents the fit function (68).
tained in [1] about the statistics of first return times. In particular, we have proved that the
distributions of the number of visits for a domain A that intersects the boundary between two
invariant regions is a linear superposition of the distributions characteristic of these regions,
weighted by coefficients equal to the relative size of the intersection of A with each invariant
region. Under a condition of continuity this also holds in the limit µ(A) → 0, when A shrinks
around a point of the boundary. Such a result allows to understand why, by coupling a regular
and a mixing system together (whose distributions follow a polynomial and an exponential
decay respectively), the limit distributions for domains containing the boundary are simply
ruled by the ones of the mixing region, although as long as the measure of the domain has
a finite positive value the regular region asymptotically gives the main contribution, which
appears like a power law tail.
As an application to a system of higher physical relevance in which regular and chaotic
motions can coexist, we have studied the distributions of the number of visits for the standard
map. The numerical analysis performed for domains wholly contained in the regular region has
shown that, as it happens for the simple model represented by our skew map, the distributions
asymptotically follow a power law decay with an exponent near 2, and there is some evidence
suggesting that the exponent slightly grows as the order k increases. For domains lying on
the chaotic sea and far away from the integrable region, the distributions depart from the
30
expected Poissonian behaviour and still decay like α t−β, with β ≃ 2. We think this fact could
be explained by considering that most of the orbits originating from points of the chaotic sea
closely approach, sooner or later, the regular region. The distributions of the number of visits
therefore provide a statistical tool capable to capture some of the fundamental features of
the dynamics. Moreover, the theoretical results obtained for systems composed by invariant
regions allow to predict and to understand the behaviour of the distributions computed for
domains which are at the boundary between the integrable orbits and the chaotic sea, as clearly
pointed out by the numerical analysis.
Another system of particular interest that we have investigated is the dissipative Henon
map. In this respect, the distributions computed for little balls centered around generic points
agree very well with the Poisson law, as usually happens for systems having a strong mixing
dynamics, and this is quite interesting since the Henon map is not uniformly hyperbolic. On the
other hand, taking little balls centered around periodic points, the distributions of the number
of visits do not show anymore a Poissonian behaviour. In fact, in this case the statistics of first
return times F0(t) is exponential with parameter less than one, which seems to depend on the
period in a straightforward way, while the distributions of order k ≥ 1 follow a different law
which is obtained from the corresponding statistics of first return times by supposing that the
differences between successive return times are independent. However, since an aperiodic orbit
can be thought like a periodic orbit with an infinite period, this law approaches the Poisson
distribution as the period grows. We think these results are in agreement with the fact that
the Henon map could enjoy a rapid decay of the correlations.
Acknowledgments
Work partially supported by the Italian M. I. U. R. project (COFIN 2003) Order and Chaos
in Nonlinear Extended Systems: Coherent Structures, Weak Stochasticity and Anomalous
Transport.
31
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33
List of Figures
1 Distributions of the number of visits Fk,A20(t) of order k = 1, 2, 3. . . . . . . . 10
2 Distributions of the number of visits Fk,A20(t) of order k = 3, 4, 5. . . . . . . . 10
3 Distribution of the number of visits of order k = 1 computed for the domain
Aǫ = [0, ǫ] × [0, ǫ] with ǫ = 10−2. The dotted line represents the function (??). 15
4 Distribution of the number of visits of order k = 2 computed for the domain
Aǫ = [0, ǫ] × [0, ǫ] with ǫ = 10−2. The dashed line represents the function 1/2 t−2. 15
5 Distributions of the number of visits of order k = 1, 2, 5, 10, 20 computed for the
domain A = [0, ǫ] × [y0, y0 + ǫ] with y0 = 0.35 and ǫ = 10−2. The dashed line
represents the function 1/2 t−2. . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6 Distributions of the best-fit parameter β obtained through a least-squares method,
in the range 30 ≤ t ≤ 100, from the distributions of the number of visits of or-
der k = 1, 2, 3. For each k, the distributions of the number of visits have been
computed for twenty domains of side ǫ = 10−2. The arrows show the position
of the mean value of β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7 Distributions of the number of visits of order k = 1, 2 computed for the map
given by Eq. (??) in the interval A = [−ǫ, ǫ] with ǫ = 5×10−4. The dotted lines
represent the theoretical predictions; in particular the one for k = 1 corresponds
to formula (??). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
8 Distributions of the number of visits of order k = 1 computed for a square
domain of side (a) ǫ = 0.2 and (b) ǫ = 0.05, such that λ = 0.5. The dotted lines
represent Eq. (??) with α = 0.5 and β = 1.94. Note that as the domain’s size
decreases, the contribution of the regular component begins to prevail for larger
values of t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
34
9 Distributions of the number of visits of order k = 1 (lower curve), k = 2 (middle
curve) and k = 3 (higher curve), computed for a square domain of side ǫ =
2.5 × 10−2 centered at (0.15, 0.25), which is wholly contained in the regular
region. The distributions appear to follow asymptotically a power law decay
like α t−β, where the least-square fit estimate of β, performed for t in the range
[10, 500], is 1.91 for k = 1, 1.95 for k = 2 and 1.97 for k = 3. . . . . . . . . . . 24
10 Distributions of the number of visits of order k = 1, 2 computed for a square
domain of side ǫ = 5× 10−2 centered at (0.7, 0.2), which lies on the chaotic sea.
The dotted line represents the function α t−β, with α = 10−2 and β = 2. . . . . 24
11 Distribution of the number of visits of order k = 3 for a square domain of
side ǫ = 0.1 centered at (0.03, 0.3), which intersects the boundary between the
regular and the chaotic regions. The dotted line represents Eq. (??) with β = 2.04. 26
12 Distributions of the number of visits of order k = 0, 1, 2 computed for a circular
domain of radius 5 × 10−3 around a generic point. The dotted lines represent
the corresponding Poisson laws. . . . . . . . . . . . . . . . . . . . . . . . . . . 27
13 Distributions of the number of visits of order k = 5, 15, 30 computed for a
circular domain of radius 5 × 10−3 around a generic point. The dotted lines
represent the corresponding Poisson laws. . . . . . . . . . . . . . . . . . . . . . 27
14 Distribution of the number of visits of order k = 0 computed for a circular
domain of radius 5 × 10−3 centered around a periodic point of period two. The
dotted line (a) corresponds to the exponential function e−t, while the line (b)
represents the function α e−αt with α ≃ 0.66. . . . . . . . . . . . . . . . . . . . 29
15 Distributions of the number of visits of order k = 1, 5, 15, 30 computed for a
circular domain of radius 5 × 10−3 centered around a periodic point of period
two. The dotted lines represent the corresponding theoretical predictions (??). 29
16 Values of the coefficient α of the distributions F0(t) computed for periodic points
of period P . The dotted line represents the fit function (??). . . . . . . . . . . 30
35