cycles of spatial discretizations of shadowing dynamical systems

Math. Nachr. 171 (1995) 95- 110

Cycles of Spatial Discretizations of Shadowing Dynamical Systems

By PHIL DIAMOND of Queensland, PETER KLOEDEN of Geelong and ALEKSEJ POKROVSKII of Queensland

(Received September 3, 1993) (Revised Version March 29, 1994)

1. Introduction

A mapping f : s1 + s2 of a metric space C? into itself is called a dynamical system. A great deal of attention has been given to systems which behave chaotically. Although there are a number of different definitions of chaos, there is general agreement among them that a number of properties seem to be common to all chaotic systems. One such is that typical trajectories o f f are bounded but not attracted to any periodic or almost periodic orbit. Another attribute is the existence of an invariant measure p whose support either coincides with s2 or has a Cantor structure, and p has distinctive mixing properties on this support.

A spatial discretization of D is a finite set L c D, and a discretization o f f is a map cp,:L -+ L whose graph Gr (cp) is close to Gr ( f ) . A system is said to have the shadowing property if any discretized trajectory off is close to some theoretical trajectory off. Many important chaotic systems are shadowing [ 101.

The question arises as to the relationship between a chaotic dynamical system f and the behaviour of discretizations of f . A basic problem here is that any trajectory of the discretization is eventually periodic, in stark contrast to the first chaotic property mentioned above. One way of resolving this apparent contradiction is to study an associated Markov chain on L [3], [9], [ll], [14]. This approach is fruitful but, from a practical point of view, it does not accurately represent what is occurring in a computer model that is a deterministic process on a finite number of states.

Another strategy, and the one used in this paper, is to consider a sequence cp, = cpLV, v = 1,2, ..., of discretizations of f on L,, where cpv -+ f and L, --f s2 in some sense as v -+ co (see [3]). Usually, the sequence of discrete spaces {L,} is thought of in terms of a sequence of discretization of s2 whose space steps h(L,) -+ 0 as v -+ co.

It is possible to characterize limits of sequences of cycles of such discretizations in terms of maximal indecomposable components of chain recurrent points off when f is continuous. I f f is not continuous, these components are replaced by equivalence classes of weakly

This research was funded under Australian Research Council grant A891 32609.

96 Math. Nachr. 171 (1995)

chain recurrent points. The substance of Theorem 2 is that if f is a shadowing map and (qY> a sequence of discretizations converging sufficiently rapidly to f as v -+ 00, then the cycles of cp, converge to these equivalence classes.

These considerations suggest a method of numerical approximation of invariant measures of f. Suppose that cp : L --t L is a discretization of f on L. The distribution of cycles of cp in L is used to approximate the invariant measure of f on a. Theorem 3 shows that all ergodic invariant measures off can be approximated in this way, by appropriate sequences of discretizations cp,: L, + L,, v = 1, 2, ... .

However, even in quite simple maps, for example f ( x ) = 2x (mod l), there exist sequences of discretizations (q,} whose cycles do not converge to the theoretical attractor as v -+ co. Consequently, the cp, do not have invariant measures approximating the canonical invariant measure p of a chaotic dynamical system f, no matter how close the discretization cp, is to f . In such circumstances the invariant measures of discretizations collapse onto a proper subset of the support of p. It is only when supp (p) has no proper invariant subsets that this does not occur, and then the corresponding sequence of discretizations is asymptotically well behaved.

Informally speaking, the essence of these results can be summarized as follows : 0 For any continuous dynamical system f there exists a discretization cp sufficiently close

to f such that cycles of this discretization mirror important properties of the original system like the structure of chain recurrent points and invariant measure. If the system f is discontinuous, then formally the same results hold but with chain recurrent points replaced by weakly chain recurrent and invariant measures by semi-invariant measures.

0 Let f be a shadowing system and let {cp,} be a sequence of discretizations converging to f so that the natural distance between graphs of the mappings f and cp, doesn’t exceed the space step of the lattice L,. Then the invariant measure o f f can be approximated by the density on cycles of the cp,, and these cycles converge to equivalence classes of weakly chain recurrent points.

0 There may exist discretizations which collapse, no matter how high the machine accuracy might be.

For clarity of exposition, proofs are relegated to the last Section.

2. Preliminaries

Let s2 denote a compact metric space with the metric e, let g be the totality of Bore1 subsets of D and let O,(B) be the c-neighbourhood of S for any S E g. Recall [2] that the Hausdorff separation of the set X E 93 from Y E is the value

Sep ( X , Y ) = inf { E : X c O,(Y)} ;

x ( X , Y ) = max (Sep ( X , y ) , Sep (X X ) )

for closed sets X , Y G s2 the Hausdorff metric between them is defined by

(1)

(A) Weakly chain recurrent points

For each E > 0 define the map j i : SZ I-+ a by f,(x) = O&(,f(x)) and introduce the sets c o r n

M , ( ~ ) = n u f : ( ~ , ( ~ ) ) , Q , wx) = n MA^). k = l n = k & > O

Diamond/Kloeden/Pokrovskii, Spatial Discretizations of Shadowing Dynamical Systems 97

A point x E SZ is called weakly chain recurrent if x E M(x). A finite sequence xl, ..., xp is called an &-cycle for f if

e(f(xi),xi+d I 8, i = 1, ..., P - 1, and e(f(xp),xl) I&. Informally speaking, a point x is weakly chain recurrent if it can be approximated by elements of &-cycles of the system f for any positive E. In contrast, a point x is called chain recurrent [lo] if this is an element of a &-cycle for any positive E . The difference is essential for discontinuous f. For instance, consider the set SZ = [0, 11 with f(x) = x2 for x < 1 and f(1) = 0. Then only the point 0 is chain recurrent, but both points 0 and 1 satisfy x E M ( x ) . Weakly chain recurrent points x, y E 51 are called connected if x E M(y), y E Mfx). The set C , ( f ) of weakly chain recurrent points is divided into equivalence classes E of points connected to each other. As noted in [8], the weakly chain recurrent property is distinct from that of nonwandering points and more general than notions of chain recurrence [lo] and first order prolongation [l]. For example, all nonwandering points are weakly chain recurrent, but not conversely. See [8] for further details. The reason for introducing this concept here is that the idea allows perturbations at each iteration of a function and this feature is essential when discretization is involved at each call of the function.

(B) Shadowing systems

A sequence {xi} of points in 0, satisfying the inclusions xi+l E &(xi) is called a 6-pseudo-orbit off . An actual orbit {y,} in SZ is said to be an &-shadow of the pseudo-orbit {xi} if e(xi,yi) < E for all i. The continuous system f is said to be shadowing [15] on a closed invariant set .4 E SZ if for every E > 0 there exists 6 > 0, such that every 6-pseudo-orbit in ( A ) is &-shadowed by the actual orbit of some yo E O,(A).

Many chaotic systems are shadowing. The classical examples are hyperbolic diffeomor- phisms and expanding transformations (see [ll, 151). Other striking examples of shadowing and non-shadowing mappings may be found in [6].

(C) Spatial discretizations

Let L be a finite subset of 52. The quantity

h(L) = sup inf {e(<, x) : 5 E L, x E SZ} x r

is called the discretization parameter on the space step of L. Let P, : SZ --+ L be a discretization operator, which is a projection, PL(x) = x if and only if x E Q [17]. Among all such we particularly identify those Pt, where Pt(x) is one of the (possibly multiple) points x, E L which is nearest in the metric e to x E SZ. So Pc could correspond to either the roundoff or truncation operator in computer arithmetic. A spatial discretization of the system f on L is a discrete dynamical system

(PL = P L 0 f : L H L ,

and (pL is a discretized dynamical system. The accuracy of approximation of such a discretization (pL is estimated by the value

4% f) = Sep (Wcp), Wf)) 3

7 Math. Nachr.. Bd. 171

98 Math. Nachr. 171 (1995)

where here Sep ( X , Y ) is the Hausdorff separation in Q x Q with expect to the metric in the space SZ x Q defined by

d(x1 , XJ3 (Yl , Y2)) = max W l ? Yl), e(x29 Y,)} .

3. @-cycles of dynamical systems

Let {L,} be a sequence of finite subsets of Q with the sequence of space steps h(L,) ---t 0 as v -, co. For brevity, write cp, = cpL, for the corresponding discretizations of f on L,, v = 1,2, ..., and P, for the discretization operators. We will consider asymptotic properties of sequences

(2)

of discretizations (p, o f f defined on a fixed sequence {L,} of discretizations of 0. If @ satisfies (2), we call it an approximating family for f . A finite sequence tl, . . . , t p E L, is called a cycle for cp, if

@ = {cpl, cp2, ...I 9 d(cp,, f ) + 0 3

cp(xi) = x i + l , i = 1, . . . , p - 1 and cp(x,) = x1

Denote by X(cp,) the totality of cycles of a discretization (p, and let X(@) denote the set of all limit points of sequences { K , } , K , E X(cp,), convergence being taken with respect to the topology of the Hausdorff metric (1). Elements of the set X(@) are called @-cycles. Standard limit constructions show that for any approximating family @ = { cpl, cpz, . . .} each @-cycle is contained in some equivalence class E of C , ( f ) . By Theorem 2 of [8] a partially converse is true:

Theorem 1. For each equivalence class E E C,( f ) there exists an approximating family @

Choose a sequence of positive real numbers r = {y,} with the following property:

for f such that E E X(@).

(3) y, 2 h(L , ) , y , + 0 as v + 00.

The y , may be regarded as bounds for the approximation errors of the discretizations (p,

from f . A sequence r satisfying (3) will be called an error sequence. Denote by X( f , r) the union of all sets X(@) corresponding to all approximating families @ = {q,} satisfying d(cp,, f ) I y,, cp, E @, v = 1,2, ... . That is, X( f , r) is the set of all possible @-cycles which are limit points of the class of sequences of spatial discretizations (9,) with one and the same error sequence { y , } , d(cp,, f ) 5 y, + 0 as v + co. For both theoretical and practical reasons, it is important to characterize the set X ( f , r) corresponding to a given error sequence r. Denote by r* the particular error sequence given by y: = h(L,).

Theorem 2. Let the continuous dynamical system f be shadowing on a closed invariant set A c Q and suppose that X is a @-cyclefor some approximating family @, satisfying X E A. Then X E X ( f , f*). In particular, if the equivalence class E is an equivalence class for the restriction f A of the system f on A then E E X( f , r*).

Theorems 1 and 2 can be extended to whole families 9 = F(f) of sequence of discretizations of a given system f . Theorem 1 relates to the family F.+ of all convergent


sequences of discretizations, while Theorem 2 refers to the class Fa of all discretization sequences {q,} satisfying e(Gr (cp,), Gr (fl) s ah(L,), where a > 0.

Theorem 2 may be applied to the investigation of robustness of attractors of dynamical systems [7], [8].

Example. Informally speaking, Theorem 2 means that for a shadowing system f , for a given equivalence class E of chain recurrent points of this system and for any sufficiently fine lattice L there exists a cycle of a natural discretization cp o f f , which approximates E in the Hausdorff distance. For nonshadowing systems, Theorem 2 fails in the following sense: for any c1 > 0 there exists a nonshadowing system f , with a corresponding equivalence class E, and a sequence of lattices L,, v = 1,2, ..., such that

E B XU, W ) where 0) = {y(4,> = {ah&)} .

That is, the cycle E cannot be approximated by natural discretizations o f f on the lattices L,.

We construct a example of such a mapping. Let 52 c lR2 be the set consisting of the segment I with end points (O,O), (1,O) and of N > CI curves w,, which are defined by y = n sin (nx), 0 x I 1, n = 1, ..., N . Define the mapping f : Q w 52 by

N The set 52 = u wn u I forms a single equivalence class for f . Note that f is not shadowing:

for each 6, a 6-pseudo-orbit n = 1

(0,0),(0,6),(0,26),...,(0,k6), k i h - ' ,

is not shadowed by any true orbit, because f is the identity mapping on I . Choose as L,, v = 1,2, . . . , any lattices of 52 satisfying h(L,) l /v with the following additional property: the restriction of L, on I is the uniform l/v-lattice consisting of the points (O,O),

Consider discretizations cp,, and cycles K , of these discretizations, satisfying Sep (52, K,) < 1/3. Such cp, and K , exist for all sufficiently large v by Theorem 1. We will show that

( m , O), ..., ( L O ) .

( 5 )

Indeed, since Sep (a, K , ) < 1/3, the cycle K , must pass along each curve w,, n = 1, ..., N . On the other hand, by the definition (4) off this cycle can go along each of these curves only in one direction, from the point (1,O) to the point (0,O). Therefore, for each sufficiently large v, the cycle K , must pass at least N times along the segment I from the point (0,O) to the point (1,O). This means, in turn, that for at least one pass along this segment the cycle must jump over at least N - 1 points of the lattice L, at one iterative step. That is, for sufficiently large v the inequality (5) holds, because the restriction of L, on I is the uniform l / v lattice and the restriction off on I is the identity mapping.

d(cp, f ) 2 N / v > ah(L,) for sufficiently large v .

7*

100 Math. Nachr. 171 (1995)

4. Cycles of spatial discretizations and invariant measures

Recall [5] that a measure p is said to be an invariant measure for f if

p(S) = p(f - ' ( S ) ) for all S E a . An important question in computer simulation of chaotic systems is as to how a good approximation for interesting invariant measures off (see [16]) can be constructed from a discretization cp. A natural approach to this problem is as follows. Having found all the cycles of a discretized approximation cp, place a uniform distribution on the points of all these cycles and use this as an approximation for an invariant measure of the system f . The validity of this method is discussed below.

For any finite subset S E SZ denote by ps the uniform probability measure

where # ( S ) denotes the cardinality of the set S . Recall [13], p. 263, that a sequence of measures {p,} is said to converge weakly to a probability measure p* if

lim j adp, = f adp , v-+m R R

for any continuous function ct : SZ -+ %. Suppose that the sequence {L,} is fixed. For each map cp,: L, H L, define the set A(cp,)

of all probability measures pK, K E X(cp,). For any approximating family of discretizations @ = {q,,}, denote by A(@) the set of all weak limit points of the sequences {p,} of elements of p, E A(cp,). The set A(@) is nonempty because each sequence of probability measures on SZpossesses a weakly convergent subsequence [13], Proposition 52.1. In order to formulate an analogue of Theorem 1, we recall two further definitions. Denote by Gr c f ) the closure of Gr ( f ) and, for any set S E a, write

f-'(S) = {x E S Z : there exists y E 3 such that (x, y ) E Gr (f)) . A probability measure p on SZ is said to be semi-invariant with respect to f if, for any Bore1 set S,

P(S) 5 P ( F ( S ) ) .

An invariant measure p is called ergodic if p(S) is equal to either 0 or 1 for any invariant subset S G SZ. Every continuous system f has at least one ergodic invariant measure [12]. Any invariant measure may be represented by a spectral decomposition into ergodic invariant measures [5].

Theorem 3. (a) Let @ be an approximating family for f . Then measures from A(@) are semi-invariant for f:

(b) For any ergodic invariant measure p.,, off there exists an approximating family @ such that p* E A(@).

Note that among elements of the set .A(@) there may appear nonergodic measures even i f f is continuous and a(?,, f ) + 0. If, for instance, f (x) = x and SZ is connected, then a probability measure p belongs to A(@) if and only if its support is connected. Recall [13]


that the support of a probability measure p is the minimal closed set S E SZ with p ( S ) = 1. On the other hand, only atomic measures concentrated at a single point are ergodic.

Because of Theorem4, it is natural that measures from the set A(@), where @ is an approximating family, should be called semi-ergodic measures. The support of any semi-ergodic invariant measure is contained in some @-cycle and, by Theorem 1, in some equivalence class E G C,cf). Every ergodic invariant measure is also semi-ergodic.

The principal result of this section is an analog of Theorem 3. First, it is convenient to introduce sets of measures corresponding to the sets Y(f, r). Let r = {y,} be an error sequence. Denote by A’( f , r) the union of sets A’(@) corresponding to all approximating families @ satisfying d(cp,, f ) 5 yy . In other words, Acf,r) is the set of all semi-ergodic invariant measures which are weak limits (as v -+ co) of sequences of measures {pK,} arising from cycles {XY} of spatial discretizations with approximation errors {y,,}.

Theorem 4. Let f be a continuous shadowing system on a closed invariant set A . Then any semi-ergodic invariant measure p with supp (p) E A belongs to A ( f , r*),

The supposition that f is shadowing cannot be omitted in Theorem4. This can be shown by a counterexample very similar to that at the end of Section 3.

5. Collapsing discretizations

Let j i be a fixed probability measure on 51, such that the measure of each open set is positive. Let pa be an absolutely continuous with respect to ji. That is,

p o ( X ) = 0 whenever ,ii = 0 for all X E ~ ,

[13], p. 245. Define the sequence of measures {p, ,} by

P , + ~ ( S ) =p,,(f-’(S)), S E B , n = 0 , 1 , 2 , . . . .

A probability measure p* is said to be attractive if all such sequences {p , } converge Cesaro weakly ([3]) to p*. That is, the Cesdro means

PO + P i + ... + P n - i m, = n

converge weakly to p* for all choices of absolutely continuous pa. In this context, here is an analogue of Theorem 5 of [S].

Theorem 5. Let f satisfy a Lipschitz condition

(6) eCf(x), f ( Y N 5 ne(X9 Y ) 1 x7 Y E Q 9

and let p* be an attractive measure with support supp (p*) E Q. Let $* be an invariant subset ofsupp (p*) . Then for any error sequence r = {y , } satisfying y, > (1’ + 1 + 1) h(L,), v = 1,2, . . ., there exists an approximating family @ of discretizations {qV} with d(cp,, f ) < y, such that all measures p E A(@) satisfy the inclusion supp ( p ) c j*.

So, for systems with a unique attractive measure k*, the approximation procedure described in the previous section must converge to just this measure and no other for any

102 Math. Nachr. 171 (1995)

sequence of discretizations, provided that its support supp (p*) has no invariant proper subsets. It is well known that the situation is often very different. In particular, for wide classes of systems with chaotic behavior there exists an attractive measure which is absolutely continuous on all of s2, in the case of expansive systems, or with support in a Cantor-like set for hyperbolic systems [16]. However, the support of the attractive measure of systems like these do contain periodic points. Hence there are invariant proper subsets consisting of cycles in supp ( p J

6. Proofs

6.1. Proof of Theorem 2

Lemma 1. Let xo, xl, x2, ..., be a trajectory o f j ; that is, x i f l = f ( x i ) , i = 0, 1, 2, ... . h(L,), a cycle Then for any v there exists a discretization 'p,:L, H L , with d(cp,, f )

K , = {yli, ..., qt)(")} E X'(cp,) and a number N ( v ) such that

@(xN(v)+i - I>d) I h(L , ) , i = 1, ..., p ( v ) .

Proof. For any fixed v , consider the sequence 5; = Ptx, , i = 0,1,2, ..., where the pro- jections P: map any point x E s2 to a point in L , which is of minimum distance fromx. Choose positive integers N ( v ) and p ( v ) such that

5;'") = ~ : ( v ) + P ( v ) , p " v ) + j + C ~ ( V ) , j = 1, ..., p ( v ) - 1 .

Define q{ = t : ( ' )+ j - ' , j = 1, ..., p ( v ) , and define the discretization (p, by

q{+l , %(d) = cp"(ul;'"') = d >

j = I, ..., p(v ) - 1 , 1 (~"(5) = P:f (5) for all other 5 E L, . The constructed objects have the necessary properties. 0

Let us complete the proof of Theorem 2. Let X c A be a given @-cycle for approximating family @, It is sufficient, for any positive E* and any sufficiently large v , to construct a discretization 'py and a cycle K , E .X(y,) such that

(7)

and d(cp,, f ) 5 h(L,). Fix E E (0, E*) and choose a 6 > 0 which figures in the definition of shadowing property.

First, construct a natural number p and a finite sequence Y = {yl, . .. , y,} E s2 satisfying the inequalities

(8)

the inclusions

Sep (K , X ) < & * , Sep ( X , K,) < E ,

Sep ( Y , X ) < + ( E , - E ) , Sep ( X , Y ) < 4 (E* - E ) ,

y j + l E f i ( y j ) , j = 1 , . . . , ~ - 1, Y i E f a ( y p ) ,

and the inclusions

Y , i = 1, ..., k .


Set CT = min IS, E* - E ) . By the definition of approximating family there exists a natural number v*, a mapping cp : L,, H Lv. and a cycle E = (tl, . . . , t,) of cp satisfying

d(q , f ) < 012, Sep (E, X ) < 012, Sep ( X , E ) < 012.

The inequality d(cp, f ) < 0/2 means that there exist elements yi, i = 1, ..., p , satisfying

e (Yi> t i )<a /2 , e U ( Y i ) , t i + l ) < c / 2 , i = l , . . . , p - 1 ,

d Y p 4,) < 012 9 e(f(y,), tl) < 4 2 .

and

The sequence Y = yl, ..., y, has the necessary properties. From the definition of shadowing property there exists a trajectory Z = {z,,, zl, ...) of

the system f satisfying e(ze yj) < E, i = j(mod p ) . In particular,

(9)

By Lemma 1, for any v there exists a discretization

Sep (Z, Y ) < E .

cp, : L, F+ L, with d(cp,, f) I W,) 9

and a cycle

K , = {d, ..., v?')} E .X(cp,)

and a number N(v) such that

@(ZN(v)+i-l,rl) g ~ ( L v ) , i = 1, . . .3p(v)2

and, further,

(10)

for sufficiently large v. The first of the inequalities (7) follows from (lo), (9) and the first of inequalities (8).

It remains to establish the second of the inequalities (7). Again consider a trajectory Z = z,,, zl, . . . , off satisfying e(ze yj) < E, i = j(mod p ) . For each natural number N , denote

Sep (K", 2) < t (E* - 4

zN = ( z N j zN+l, ...) Z N + p - l ) '

Clearly, for each N

(11) Sep ( Y , ZN) < E .

On the other hand, by Lemma 1 for any v there exist as a discretization

cp, : L, F+ L, with d(cp,, f) < W,) ,

a cycle

K , = {d , ..., r ~ ' " } E . f ( c p , )

and a number N(v) such that

@(ZN(")> d ) m,) '

Math. Nachr. 171 (1995) 104

By the continuity off ' there exists v* such that for v > v* the following inequalities hold:

e ( Z N ( v ) + i - l , d ) < & ( E * - 4, i = l , . . . , ~ , i =j(modp(v)),

and

(12) SeP ( Z N ( V ) , K , ) < f (E* - 4 for sufficiently large v. The second of inequalities (7) now follows from (12), (11) and the second of inequalities (8). This completes the proof of the theorem. 0


In the proof of assertion (a), the following known result will be used.

Lemma 2. Let a sequence of probability measures {p,) converge weakly to a probability measure p*. Then for any closed set S E s2,

lim PAS) P*(S) v* m

and for any open set U G s2,

lim P A W 2 P* ( W v + m

See for example [13], Proposition 51.2, of which it is part.

measures p K , converge weakly to a probability measure p*. We should establish inequality Let a sequence of discretizations (p, and a sequence K , E X(cp,) be chosen such that the

for each set S E W.

then the relations Let us prove first the inequality (13) for any closed S s 0. Let E > 0. If d(cp,,f) < E ,

5 E O,(S) n L, and 5 = ~ , ( d

4 E O e ( f - % J 2 m .

imply

Therefore, for sufficiently large v, the inequality

P K , ( o ~ ( S ) ) 5 P K , ( o o , ( f - ' ( O ~ Z ( ~ ) ) ) )

holds. Note that the set O,(S) in the left-hand side of this last inequality is open and the set O,(f- 1(02E(S))) in its right-hand side is closed. Thus, by Lemma 2,

(14) P*((",(S)) P*(~,(S-l(@ZE(S)))). Because S is closed


Therefore inequality (13) for a closed set S is the limit of (14) as E -+ 0. Let now S be an arbitrary set from a. It is well known ([13], Proposition 19.16), that

there exists a sequence of closed sets S, c S such that

(15) P*(Sfl) -+ P * N as -+ .

(16) P*(Sfl) 5 P*(fT(Sfl)) *

(17) P * ( . F ( U 5 P * ( f T ( a .

Then, as it has been established,

On the other hand,

Inequality (13) follows from (15)- (17) and the assertion (a) is established. Let us turn to the assertion (b).

Lemma 3. Let m be an ergodic invariant measure, let a1 . . . , ctN be functions in C(SZ), and let E > 0. Then there exists an integer v* > 0 such that f o r each v 2 v* there exists a discretization (p,: L, H L, and a cycle K , E X(rp,) with period p ( v ) satisfying

(19) 4 c p V 9 f ) 5 8 .

Proof. Let E > 0 be fixed. Throughout the proof we will assume that there is a fixed positive p < E such that

(20) SUP {I%(X) - a,(Y)l: @(X, Y ) < p} < E/2, k = 1, . .. N .

Such p exists because 52 is compact. Establish firstly that there exist xo E SZ and a positive integer p satisfying

and

(22) e(fp--’(xo)9 xo) < B . By the Birkhoff-Khinchin theorem [5] there exists y o E 52 such that

u,(X)dm, k = 1, ..., N , R

(23)

hold. Let y* be a limit point of the sequence yo, f ( yo ) , . . ., f”(yo), . . ., and e(fnoyo, y,) < p/2. By (23) for each t > to inequalities

106 Math. Nachr. 171 (1995)

hold. It remains to define x , = f”O(y,) and to choose as p any natural number satisfying p > 6, and eCfno+p-’y,, y,) < P/2.

(25)

Choose an integer v* > 0 such that for all v 2 v*

Denote by d the quantity

4 = min {e(f’(xo), f ” ( x o ) ) : f i ( x 0 ) =t= f j (xo) , i,j = 1, ..., p(v)} .

(26) W,) < P 3 4,) < E - P 9 m,) < $/2.

It should be emphasised here that v* depends only on the lattices L,, which are given, but does not depend on the discretizations q, which are in fact constructed in this proof.

By (21) and (22) the proof will be finished if for each v 2 v* we construct a discretization cp, satisfying (19) and a cycle K, = {t:, ..., (1) E X(cp,) satisfying

(27)

By (20) it is sufficient to construct a discretization 50, and a cycle K , satisfying

lcx,(c{) - cc,(fj(x,))l < ~ / 2 , j = 1, ..., p , k = 1, ..., N .

(28)

and this comes to the same thing as (27). Define

e ( C . f j ( x d < P , j = 1, ..., P ,

= P: fJ (xo ) , i = I, ..., p .

From the first inequality (26) the points satisfy the inequalities (28). It remains to prove the existence of a discretization (p, satisfying (19) and having the points 5; as a cycle. Without loss of generality consider only the case when all points f’(x,), i = 1, ..., p , are distinct. Then by (25) and the third inequality (26) the points (f = P,*f’(x,), i = 1, ..., p , are also distinct. By the second inequality (26) these points satisfy the estimate

e((51,5t”) ,Gr(f))<&, i = I , . . . , p - 1,

and by (22) and the second inequality (26) also the inequality

e ( (55 c:)? Gr (f)) < E .

Let us now define a discretization (p, by

( ~ ~ ( 5 9 = 4+’, j = 1, ..., P - 1, ~ ~ ( 5 1 ) = t:, and by

(p,(() = P : f ( c ) on all other points ( E L,.

By the construction, this discretization automatically satisfies (19) for v > v* and has the points as a cycle. The proof is complete. 0

Let us return to the proof of assertion (b). Let m be an ergodic invariant measure. It is sufficient to construct discretizations q, and cycles K , E X”(cp,), v = 1,2, ..., so that the sequence of measures p, = pKv is weakly convergent to m and

(29) 4% f ) + 0 .


Let ak(x) be a sequence of continuous functions dense in C(s2). By Lemma 3 it is possible to choose an increasing sequence v ( k ) such that for each v 2 v(k) , k = 1,2, . . . , there exists a discretization cp': : L, H L, and a cycle C{ E X(q{) of period p{ satisfying

The inequalities (30) may

I " be rewritten in the form

...,

...,

k ,

k .

Put

k(v) = max { k : v ( k ) I v} , v = 1, 2, . .. ,

and define

(p, = (pk,"', K , = Ck,"'.

By (31) the relation (29) holds. It remains to prove that the sequence py = pKv is weakly convergent to m. Measures p, are probability measures, in particular, the set {p,, v = 1,2, . . .) is uniformly bounded in the Banach space of all finite Bore1 measures on s2 [13]. Therefore, by virtue of (32).

j .(x) dp, -+ a(x) dm as v -+ co R R

(33)

for each limit point a(x) of the sequence aj. Due to density of the sequence cl j , it means that the measures p, converge weakly to m. This completes the proof of Theorem 3. 0


Lemma 4. Suppose that the continuous system f is shadowing. Let m be an ergodic invariant measure, let alt ..., aN be continuous functions on SZ and let c > 0. Then for each sufficiently large v there exists a discretization (p,: L, H L, and a cycle K , E X(cpv) with period p(v) satisfying the inequalities

(35) 4% f ) 5 h(L,).

(Compare the inequality (35) with the inequality (19)).

108 Math. Nachr. 171 (1995)

Proof. Let E > 0 be fixed. Throughout the proof we will assume that P > 0 is a fixed positive constant such that

SUP { h ( X ) - %(Y)1: @(x9 Y) < p } < &/4 , k = 1, . . . , N .

First of all, for such P, it is possible to find a value 6 appearing in the definition of shadowing property. Without loss of generality we may assume that 6 I 8. By Lemma 3 there exists an integer v*, a discretization 'p* : Lv, H L,* and a cycle

of period p* , satisfying

and

From (36) it follows that there exist elements y i E D, i = 1, ..., p* , satisfying e(q i , t i ) < 6 and

(38) q y ' ~ o ~ ( f ( & ) ) , i = 1 ,..., p* - 1, s: E o"a(?",*)).

(39) e ( q : , t i * ) I P , i = L . . . , p * .

Since 6 I fl,

By the definition ofthe shadowing property and (38), there exists a trajectory Z = {zo, zl, . . .> of the system f which satisfies the inequality

(40) e(z,, q:) < P , i = 1, ..., P* 3

Taking (39) and (40) gives

(41) e(z,,<:) < 2 P , i = 1, ..., p * , n = i(mod p,) .

It follows from (37) and (41) that there exists a positive integer &* such that for each natural number no and each L 2 t?, the inequality

n = i(mod p,) .

holds. Then by Lemma 1 for each v there exists a system qV : L, H L, satisfying (35) and a cycle K , = { q i , ..., i f ( ' ) } E X(cp,), such that

(43)

Choose a natural number q satisfying q > L, /p(v) . By the continuity off, (43) implies that

(44)

e ( z n 0 ( v ) 9 YI3 < P >

e(znOc,, tj- I, qt) < B , j = 1, . . . , qp(v) , j = i(mod ~ ( v ) ) ,


for sufficiently large v. From (42) and (44) it follows that

for sufficiently large v. That is, the cycle K , satisfies (34), and Lemma 4 is proved. 0

Theorem 4 now follows from Lemma 4 in much the same way as the statement (b) of Theorem 3 was obtained from Lemma 3. This completes the proof of the theorem. 0


Denote the support of p* by gP*. Introduce the notations

Yo(&) = O,(Y*) and

$k(E) = f - ' $ k I - t ( & ) , k = 1, 2, ... . Further, put

Q k ( E , 6) = O a ( $ k ( E ) ) , k = 0, 1, 2, ... .

By the definition of attractive measure for any E , 6 > 0 the union of sets Q k ( & , S), k = 0, 1,2, ... , coincides with s1. For any positive integer v, let

O0(&, 6, v) = Qo(&, 6) n L , ,

@k(&, 6, V ) = ( Q k ( 4 6) \ Q k - i ( & , 6)) n L,, k = 1, 2, ... . Define the map ( ~ ( 5 ; E, 6, v) by

( ~ ( 5 ; E, 6, V ) = P:f(Tk(&) 51, 5 E @ k ( & , 6, v ) , k = 0, 1, ... ,

where the mapping P,* is as defined in the proof of Theorem 2, and K(E) 5 is the nearest of the points of $k(&) to the point 5 E L(v). For 6 > h(L,) the inclusion

q ( @ k ( & , 6, V); E , 6, V) 5 @ , - I ( & , 6, V), k = 0, 1, ... ,

is clearly true. Due to the Lipschitz condition (6), if 6 > 1 ( ~ + h(Lv)), the set O0(&, 6, v) is invariant for the map ( ~ ( 5 ; E, 6, v). Finally, for 61 + h(L,) < y,, the map ( ~ ( 5 ; E, 6, v) satisfies the estimate d(q, f ) < yv . Thus, if y v satisfies

Y Y > ( A 2 + 1 + 1 ) W V ) ,

then, for sufficiently small positive E = ~ ( v ) and for 6 = 6(v) = (1 + 1) h(L,), it follows that the function q(5; ~ ( v ) , 6(v), v) satisfies d ( q , f) < y Y . Then every cycle of q(5; ~ ( v ) , S(v), v) is contained in the ( ~ ( v ) + h(Lv))-neighborhood of the set gP*, and the proof of Theorem 6 is complete. 0

110 Math. Nachr. 171 (1995)

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Mathematic Department University of Queensland 4072 Australia

Current Address: Mathematics Depar tmen t University of Queensland 4072 Australia

Department of Mathemutics Deski University Geelong 321 7 Australia

Permanent Address: Institute of Information Transmission Problems Russian Academy of Science Moscow

cycles of spatial discretizations of shadowing dynamical systems

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