SPECTRA OF DIMENSIONS FOR POINCAR ´ E RECURRENCES FOR SPECIAL FLOWS

Share Embed


Descrição do Produto

TAIWANESE JOURNAL OF MATHEMATICS Vol. 6, No. 2, pp. 269-285, June 2002 This paper is available online at http://www.math.nthu.edu.tw/tjm/

SPECTRA OF DIMENSIONS FOR POINCARE´ RECURRENCES FOR SPECIAL FLOWS V. Afraimovich, J.-R. Chazottes and E. Ugalde

Abstract. We prove the variational principle for dimensions for Poincar ´e recurrences, in the case of invariant sets of dynamical systems with continuous time. To achieve this goal we show that these dimensions can be expressed as roots of a non–homogeneous Bowen equation.

1. INTRODUCTION Typical motions in dynamical systems repeat their behavior in time. Complexity or simplicity of dynamics depends on this repetition, and quantitative properties of behavior of orbits can be often expressed in terms of Poincare´ recurrences. Of course, it is generally impossible to describe completely the return behavior of every orbit. Traditionally people study statistical properties of Poincare´ recurrences (see, for instance [13]). In this approach one does not take into account sets of zero measure which can be very large when measured by means of Hausdorff dimension or topological entropy [3]. A new approach has been proposed recently [2, 4] that makes use of ideas and methods of dimension theory [9]. The spectra of dimensions for Poincar´e recurrences has been introduced and studied [1, 2]. It was shown in [2] that for maps on conformal zero-dimensional repellers, the spectrum of dimensions is defined by roots of a non-homogeneous Bowen equation. A similar problem for systems with continuous time was unsolved. While introducing a dimension-like characteristic of sets, one can immediately introduce a dimension for the measure as follows (see [9] for a more general definition). Received December 12, 2001. Communicated by S. B. Hsu. 2000 Mathematics Subject Classification: 37C45. Key words and phrases: Dimension theory, Poincar´e recurrences, special flows.

269

270

V. Afraimovich, J.-R. Chazottes and E. Ugalde

Let F be a collection of subsets of a metric space (X; d), such that for any ² > 0 there exists finite or countable subcollection G ½ F, of subsets with diameter less than or equal to ², which is a cover of X. Let », ´ : F ! R+ be functions such that »(;) = ´(;) = 0, and ´(U) = 0 iff U = ;. Given Z ½ X, ²; ® > 0, form the partition function 0 1 X M(Z; ® ) ´ lim @ inf »(Bi )(´(B i))® A : ²!0

diam(G)· ² B 2G i

where the infimum is take over finite or countable covers from F of the set Z. It was shown in [9] that there exists a critical value ® = ® c 2 [¡ 1; 1], such that M(Z; ®) = 0 if ® > ® c , and M(Z; ®) = 1 if ® < ® c. The value ® c is called the Carath ´eodory dimension of the set Z, corresponding to the Carath ´eodory structure (F; »; ´). We will say that ¯ is a dimension-like characteristic if ¯ = ® c for some Carath e´odory structure (F; »; ´). When one has a dimension-like characteristic ¯, and a probability measure ¹ on X, one may introduce the dimension of measure dim¯ (¹) as follows dim ¯ (¹) = inff¯(Z) : ¹(Z) = 1g: Two problems appear to be considered. (1) Variational principle: is it true that supfdim¯ (¹)g = ¯(X ) where supremum is taken over all Borel probability measures? (2) Existence of measures of full dimension: is there at least one Borel probability measure ¹ 0 such that dim¯ (¹ 0) = ¯(X)?

In the present work, in the capacity of ¯ we consider dimensions for Poincare´ recurrences (see below). Let us notice that for this dimension-like characteristic the first problem was solved for maps acting on Cantor sets resulting from Moran type constructions (conformal zero-dimensional repellers) [2]. The authors made use of non-homogeneous Bowen equation. In the present article we prove a similar result for a class of systems with continuous time, namely suspension flows on conformal repellers maps. (Let us recall that special flows over subshifts of finite type are the basic ingredient in the construction of the symbolic dynamics of hyperbolic and Anosov flows [5].) Our strategy is also the derivation of a non-homogeneous Bowen equation for special flows, and then to show that spectra for Poincar´e recurrences for flows are determined by roots of this

Spectra of Dimensions for Poincar´e Recurrences for Special Flows

271

equation. This, together with known results for maps in [4], lead to the variational principle for dimensions for Poincare´ recurrences, the main result of this article. Let us notice that asymptotic properties of Poincare´ recurrences are not so well studied and understood. The main classical result concerns the distribution of periodic orbits (zeta functions); see for instance [8]. Therefore our results and the ones in [4, 7] provide a new insight into the nature of recurrences for special flows. As far as the above problem (2) is concerned, the problem is solved for Hausdorff dimension in the case of a class of maps on conformal repellers (see for instance [11]). For the conformal axiom A flows, this problem was solved in [10]. For dimensions of Poincare´ recurrences, however, the question remains open. In Section 2 we list notions and results needed for the formulation and the proof of the main result. In Section 3 we formulate and prove the variational principle for special flows over conformal zero-dimensional repellers. 2. PRELIMINARIES 2.1. Maps under consideration Following the definitions and notation in [4], let (X; T)W be a subshift of (§ N ; T ). 1 ¡j n Denote ³ the partition of X into 1-cylinders, and ³ = n¡ ³, n ¸ 1, the j=0 T 0 dynamical refinements of this partition.S Finally, let us denote ³ the trivial partition n + fX; ;g. The length of a cylinder ¢ 2 1 n=0 ³ is the integer j¢ j = maxfm 2 Z : m ¢ 2 ³ g which coincides with the number of symbols in § needed to determine ¢. We assume that (X; T ) is weakly specified, i. e., there exists an integer n0 2 Z+ such that for any two cylinders ¢ 2 ³n and ¢ 0 2 ³m , and for each integer p ¸ n + m + n0 , there exists a periodic point x ~ of period p such that x ~ 2 ¢ and T n+n0 x ~ 2 ¢ 0 . Classical examples of specified subshifts are subshifts of finite type and their topological factors, namely sofic subshifts. For each x 2 X let ¢ n(x) be the atom of ³n containing x, i. e. ¢ n (x) is the only cylinder of length n containing x. We now define a metric on X generating the product topology. Let u : X ! (0; 1) be a continuous function and define the metric (1)

n (u(¢ n(x)))

dX (x; y) = e¡ S

P 1 where n = supfk 2 N : ¢ k (x) = ¢ k (y)g and S n (u(¢ n (x))) = maxz2¢ n (x) n¡ j=0 u(T j z): In the sequel we will assume that u is Ho¨lder continuous. It is not hard to see that dX an ultrametric. Furthermore, open balls coincides with cylinder sets, i. e., for each x 2 X and ² > 0 there exists a unique n(x; ²) 2 Z+ such that B(x; ²) = ¢ nx;² (x). And vice versa, for each x 2 X and n 2 N there

272

V. Afraimovich, J.-R. Chazottes and E. Ugalde

exists ²(x; n) > 0 such that ¢ n(x) = fy 2 X : dX (x; y) < ²(x; n)g, thought the choice of ²(x; n) is not unique. All T-invariant probability measures are assumed to have a strictly positive entropy, i.e. h¹ (T ) > 0. 2.2. Poincare´ recurrences For any set U ½ X define its Poincare´ recurrence as ¿T (U) = minfk 2 N : T k (U ) \ U 6= ;g: Note that ¿T (U) = ¿T (T ¡ 1 U) for any U ½ X, and that ¿T (A) · ¿T (B) whenever A ½ B. 2.3. Special space and special flow Let Á : X ! (0; 1) be a Ho¨lder continuous function. Consider the interval [0; maxx2X Á(x)] with the usual topology, and endow X £ [0; maxx2X Á(x)] with the product topology. The special space defined by Á over X is the quotient space X Á ´ f(x; t) : x 2 X; t 2 [0; Á(x))g; defined by the quotient map (x; t) 7!

½

(x; t) if t < Á(x) (Tx; 0) if t ¸ Á(x):

To simplify the notation let (x; tÁ(x)) be denoted by xt. In the special space X Á we define the special flow © : X Á £ R+ ! X Á by ½ xs+t if s + t < 1 © (xs ; tÁ(x)) = (T n (x))s0 if s + t ¸ 1; where n = minfk 2 N :

k P

j=0

j

0

Á(T (x)) ¸ (s+t)Á(x)g and s =

(s+t)Á(x)¡

nP ¡1

Á (T j (x))

j =0 Á (T n (x))

2.4. Bowen-Walters’distance For Á = 1 there exists a natural metric generating the quotient topology on X 1, which was first introduced in [6]. This definition can be readily adapted for the general case. Consider the t-horizontal sections Xt ´ fxt 2 X Á : x 2 Xg, and define the t-horizontal distance ½t (xt ; yt ) = (1 ¡ t)dX (x; y) + tdX (Tx; T y).

:

273

Spectra of Dimensions for Poincar´e Recurrences for Special Flows (0)

(1)

(n¡ 1)

A path p between xt and yt 0 is a finite sequence p = fxt = xt0 ; xt1 ; : : : ; xtn¡ 1 ,

(n)

xtn = yt0 g, such that for each 0 · i < n, if x(i+1) 62 fx(i) ; T x(i) g then ti = ti+1. Pn¡ 1 (i) (i+1) The length of the path p is given by jpj ´ i=0 jfxti ; xti+1 gj, with (i+1) jfx(i) ti ; xti+1 gj

=

(

1 ¡ ti + ti+1 (i) (i+1) ½ ti (xti ; xti ) + jti+1 ¡

if Tx(i) = x(i+1) and ti > t i+1; ti j otherwise.

Finally, the distance in the special space is given by dXÁ (xt; yt0 ) = inffjpj : p 2 [xt ! yt0 ]g; where [xt ! yt0 ] denotes the set of all paths from xt to yt0 . 2.5. Spectrum of dimensions In the computation of the Poincare´ spectrum for © , we will use covers of X Á by special open sets we call rectangles. Given ² > 0 and xs 2 X Á , define the s-horizontal open ball of radius ², S(xs ; ²) = fys 2 Xs : ½ s(xs ; ys ) < ²g. The rectangle with base S(xs ; ²) and of height ± > 0 is the set [ [ R(xs ; ²; ±) = © (ys ; Á(y)t): ys2S(xs;²) 0 0; see [4] for counter-examples. 2.7. Topological pressure

Let (X; T ) be a subshift of (§ N ; T ), and à a real-valued continuous function P Pn¡ 1on j X. Let Zn(à ; X) = ¢ 2³ n exp(Sn (à (¢ ))), where S n(à (¢ )) = supx2¢ j=0 à (T x). 1 It was proved in [12] that the limit PX (à jT ) = limn!1 n log Zn (à ; X) exists. This limit is called “the topological pressure of the function à on X with respect to T”. For every constant c 2 R, the topological pressure satisfies PX (c + ÁjT ) = c + PX (ÁjT ). There is a dimension-like definition of the topological pressure [9] based on a Carath ´eodory construction. For a countable cover C of S by cylinders and ¯ 2 R let X Z(¯; à ; C; X) = exp(¡ ¯j¢ j + à (¢ )): ¢ 2C

Spectra of Dimensions for Poincar´e Recurrences for Special Flows

275

It was proved in [9] that the topological pressure PS (Ã ) coincides with the threshold value n o PX (Ã jT ) = sup ¯ : lim (inffZ(¯; Ã ; C; X) : jCj ¸ ng) = 1 ; n!1

where jCj = n denotes the minimal length of the cylinders in C.

3. A VARIATIONAL PRINCIPLE FOR THE POINCARE´ SPECTRUM As mentioned in the introduction, the aim of this paper is to prove a variational principle for the spectrum for Poincare´ recurrences for a special flow. Our main result is the following. ¹ e be the set of all ergodic © -invariant probability measures Theorem 3.1. Let M ¹ e g. in X Á . Then, for ® ¸ 1 and q ¸ 0; ®(X Á ; q) = supf® ¹ (q) : ¹¹ 2 M The proof of this result is based on Theorem 2.1 and the Bowen-like equation stated in the next theorem. Theorem 3.2. For ® ¸ 1 and q ¸ 0; the spectrum for Poincar´e recurrences ® ¤ ´ ®(X Á ; q) satisfies the Bowen-like equation PX ((1 ¡ ® ¤ )u ¡ qÁjT ) = 0. Remark 3.1. If q = 0, we obtain the result of Theorem 4.2 in [10]. This equation follows from two inequalities. Claim 3.1. If ® ¸ 1 and q ¸ 0 are such that PX ((1 ¡ ®)u ¡ qÁjT) < 0, then ® ¸ ® ¤ ´ ® (X Á ; q). Claim 3.2. If ® ¸ 1 and q ¸ 0 are such that PX ((1 ¡ ®)u ¡ qÁjT ) > 0; then ® · ® ¤ ´ ® (X Á ; q). The first inequality is proved by exhibiting a particular sequence of covers by rectangles fRn : n 2 Ng, such that diam(Rn) ! 0 when n ! 1; and limn!1 M(X Á ; ® ; q; Rn ) < 1; whenever PX ((1 ¡ ®)u ¡ qÁjT ) < 0. For the second inequality we will require the definition of the pressure. In both cases we will need the following results. 3.1. Concerning the diameter of rectangles Lemma 3.1. Let xt ; y t0 2 X Á be such that ¢ 1(x) = ¢ 1(y); and jt ¡ t0 j + ½s (xs ; ys ) · 1 ¡ t; then dXÁ (xt ; y t0 ) = ½ s(xt ; yt) + jt0 ¡ tj; with s = min(t; t0 ).

276

V. Afraimovich, J.-R. Chazottes and E. Ugalde

Proof. Let (.z; z0 ) = dX(T z; T z0 ) ¡ dX (z; z0 ) for each z; z0 2 X. Since ½s 0 (zs0 ; zs0 0 ) = ½ s (zs ; zs0 ) + (s0 ¡ s)(.z; z0 ), then ½ s0 (zs0 ; zs0 0 ) ¸ ½s (zs ; zs0 ) whenever s < s0 and (.z; z0 ) ¸ 0. (0)

(1)

(m¡ 1)

Non-traversing and positive paths. Let q = fxt = zt0 ; zt1 ; : : : ; ztm¡ 1 ;

(i+1) = z (i) whenever z(m) tm = yt0 g 2 [xt ! yt0 ] be “non–traversing”, i. e., z (i) ti 6= ti+1. Suppose also that q is “positive”, i. e., (.z ; z (i+1) ) ¸ 0 for each 0 · i < m. Then we have

jqj ¸

m¡ X1

(i+1) ½s 0 (zs(i) )+ 0 ; zs 0

i=0

m¡ X1 i=0

jti+1 ¡ ti j

(m) 0 ¸ ½ s0 (zs(0) 0 ; zs 0 ) + jtm ¡ t0j + 2(s ¡ s ) (m) 0 0 (0) (m) )) = ½ s(z(0) s ; zs ) + (t0 + t m ¡ 2s ) + (s ¡ s )(2 ¡ (.z ; z ¸ ½ s(xs ; ys ) + jt0 ¡ tj;

where s0 = mini ti and s = min(t 0; t m). Hence, our first partial conclusion is that inffjqj : q 2 [xt ! yt0 ] non-traversing and positive g ¸ ½ s (xs ; ys ) + jt 0 ¡ tj: 1) (m) Non-traversing general paths. Let q = fxt = zt(0) ; zt(1) ; : : : ; zt(m¡ ; ztm g 2 0 1 m¡ 1 (j) (j+1) [xt ! y t0 ] be non-traversing. Suppose that (.z ; z ) < 0 for each j 2 fi; i + 1; : : : ; i + k ¡ 1g ½ f0; 1; : : : ; m ¡ 1g. Then necessarily z(j) 6= z (j+1) and tj = t i (j) (j+1) for all j 2 fi; i+ 1; : : : ; i+kg. Since ½ti (zti ; zti ) = (1¡ ti )dX (z (j) ; z(j +1) ) = 1 ¡ ti , then we have (i)

(i+1)

jfzti ; zti

(i+k)

; : : : ; zti

(i)

(i+k)

gj = (k ¡ 1)(1 ¡ ti) ¸ ½ ti (zti ; zt i

) = 1 ¡ ti

whenever (.z (i) ; z i+1 ) < 0. Hence, to each non-traversing path q, we may associate another non-traversing (0) (1) (m0¡ 1) (m0 ) path q 0 = fxt = wt0 ; wt1 ; : : : ; wtm0¡ 1 ; wt m0 = yt0 g 2 [xt ! yt0 ], with m · m0, such that if ±(w(i) ; w(i+1) ) < 0 then ±(w(i¡ 1) ; w(i) ) ¸ 0, for each 1 · i < m0. Furthermore jq0j · jqj. Now, for 1 · i < m0 such that ±(w(i) ; w(i+1) ) < 0, and for arbitrary small ´ > 0 we have (i¡ 1)

(i)

(i+1)

jfwti¡ 1 ; wti ; wti

(i¡ i)

(i)

(i¡ i)

(i)

gj = ½ ti¡ 1 (wti¡ 1 ; wti¡ 1 ) + jti ¡ ti¡ 1j + 1 ¡ t i

(i)

(i+1)

· ½ ti¡ 1 (wti¡ 1 ; yt i¡ 1 ) + jt i ¡ ti¡ 1j + ½t i¡ 1 (yti ; wt i (i¡ 1)

(i)

(i)

(i+1)

· jfwti¡ 1 ; yti¡ 1 ; yti ; wti

gj + ´;

)+ ´

Spectra of Dimensions for Poincar´e Recurrences for Special Flows

277

provided that y (i) 62 fw(i) ; Tw(i) g satisfies dX (z (i) ; y(i) ) < (1+2exp(max u))¡ 1´. Furthermore, both (.w(i) ; y(i) ) and (.y (i) ; w(i+1) ) are non-negative. By doing this replacement for each 1 · i < m such that (.w(i) ; w(i+1) ) < 0, we obtain a nontraversing chain 1) (n) p0 = fxt = yt(0) ; yt(1) ; : : : ; y(n¡ tn¡ 1 ; y tn = yt0 g 2 [x t ! yt0 ]; 0 1

such that with n < 2m0 < 2m elements. This chain is that jp0 j ¸ jq0j + m´. If in addition p0 is such that ±(y (0) ; y (1) ) ¸ 0, then p 0 is non-traversing and positive, and as we proved before, it has length bounded below by ½ s (xs ; ys )+jt0 ¡ tj. Suppose on the contrary that (.y(0) ; y (1) ) = (.x; y (1) ) < 0, then necessarily t1 = t, Tx = Ty (1) and ¢ 1(x) \ ¢ 1(y (0) ) = ;. Since y 2 ¢ 1 (x), then dX(y (1) ; y) = (1) (n¡ 1) (n) 1 > dX (x; y). Using this and the fact that fyt ; : : : ; ytn¡ 1 ; ytn = yt0 g is nontraversing and positive, we obtain 1) (n) jp0 j = (1 ¡ t) + jfyt(1) ; : : : ; y(n¡ tn¡ 1 ; yt n = yt0 gj 0 ¸ (1 ¡ t) + ½s (y(1) s ; ys ) + jt ¡ tj

= (1 ¡ t) + (1 ¡ s) + sdX (T x; Ty) + jt 0 ¡ tj ¸ (1 ¡ s)dX (x; y) + sdX (T x; T y) + jt0 ¡ tj = ½ s (xs ; ys ) + jt0 ¡ tj:

In conclusion, for each non-traversing q 2 [xt ! yt0 ], and each ´ > 0 we have jqj ¸ jt ¡ t0 j + ½s (xs ; ys ) + m´, where m is the number of elements in q. Since ´ can be taken arbitrarily small, then inffjqj : q 2 [xt ! yt0 ] non-traversingg ¸ ½s (xs ; ys ) + jt0 ¡ tj: General paths. It remains to prove that the minimal length is reached inside the (0) (1) (m¡ 1) (m) set of non-traversing paths. For this, take p = fxt = xt0 ; xt1 ; : : : ; xt m¡ 1 ; xtm = yt0 g 2 [xt ! yt0 ] such that t i 6= ti+1 and x(i+1) = Tx(i) for some i 2 f0; 1; : : : ; m¡ 1g. Let j = minfi : ti 6= ti+1 and x(i+1) = T x(i) g; (0)

(j)

so that the path fxt = xt0 ; : : : ; xtj g is non-traversing. Let s0 = min(t; tj ), then, by using jtj ¡ tj = t + t j ¡ 2s0 we have (j)

(j +1)

jpj ¸ d© (xt ; xtj ) + 1 ¡ t j + t j+1 + d© (xtj +1 ; yt0 ) (j)

¸ jt ¡ tj j + ½ s0 (xs 0 ; xs0 ) + 1 ¡ tj ¸ t ¡ s0 + 1 ¡ s0 = 1 ¡ t:

278

V. Afraimovich, J.-R. Chazottes and E. Ugalde

Since by hypothesis 1 ¡ t ¸ ½s (xs ; y s) + jt0 ¡ tj, we conclude

inffjpj : p 2 [xt ! yt0 ]g ¸ ½ s(xs ; ys ) + jt 0 ¡ tj;

whenever ½ s (xs ; ys ) + jt0 ¡ tj < 1 ¡ t. On the other hand jfxt ; xs ; ys ; yt0 g = ½s (xs ; ys ) + jt0 ¡ tj, the lemma follows. Remark 3.2. Since jt0 ¡ tj+½s (xs ; ys ) = jfxt ; xs; ys ; y t0 gj, then dXÁ (xt ; yt0 ) · jt ¡ tj + ½ s (xs ; ys ), where s = min(t; t0 ). On the other hand, if jt 0 ¡ tj > 1 ¡ jt0 ¡ tj then, using the path fxt ; y~t ; yt 0 g with y~ such that T (~ y ) = y, we obtain dXÁ (xt; yt0 ) · ½ t (xt; y~t ) +1 ¡ jt ¡ t0 j. Since the system (X; T) is assumed to have the specification property, one always has T ¡ 1(y) 6= ;. 0

The projection in 2 X of the rectangle R(xs ; ²; ±) is the set fy 2 X : ys 2 S(xs ; ²)g: This projection does not depend on ±, so that we can denote it by ¢ (xs; ²). Furthermore, ¢ (xs ; ²) is a cylinder set : Proposition 3.1. For each xs 2 X Á and ² > 0 there exists m(xs; ²) 2 N such that ¢ (xs ; ²) 2 ³m(xs ;²) . set

Proof. For every y 2 X nfxg, the distances ½s (xs; ys ) belong to the countable Dx;s ´

n o n n n¡ 1 (T x)) dx;s;n = (1 ¡ s)e¡ S (u(¢ (x)) + s e¡ u(¢ :n2N ;

with no accumulation points other than zero. Since u > 0, then dx;s;n = dx;s;m if and only if m = n. Clearly dx;s;0 > dx;s;1 > ¢¢¢ is a decreasing sequence converging to zero. Since y 2 ¢ (xs ; ²) () ½s (xs ; ys ) = dx;s;m < ², with m = maxfn 2 N : ¢ n (x) = ¢ n (y)g, then the result follows with m(xs; ²) = minfn 2 N : dx;s;n < ²g. 3.2. Concerning Poincare´ recurrences Lemma 3.2. There exists a constant a ¸ 0 such that for any xs 2 X Á and 0 < ±; ² < 1; S ¿ (Á(¢ (xs ; ²))) ¡ a · ¿© (R(xs ; ²; ±)) · S ¿ (Á(¢ (xs ; ²))) + a P where ¿ := ¿T (¢ (xs ; ²)); and S ¿ (Á(¢ (xs ; ²))) = maxz2¢ (xs;²) ¿j=0 Á(T j z).

Proof. Let ¿ = ¿T (¢ nx;² (x)). For each z 2 ¢ (xs ; ²) \ T ¡ ¿ (¢ (xs ; ²)), and each ´ 2 (0; min(±; 1 ¡ s)), if ¿

s(Á(T (z)) ¡ Á(z)) ¡ ´Á(z) < t ¡

¿¡ X1 j =0

Á(T j (z)) < (s + ´)(Á(T ¿ (z)) ¡ Á(z));

Spectra of Dimensions for Poincar´e Recurrences for Special Flows

279

then both zs+´ and © (zs+´ ; t) belong to R(xs ; ²; ±). Therefore ¿© (R(xs ; ²; ±)) ·

max

z2¢ (x s;²)

¿ X

Á(T j z) + 2max Á:

j=0

In this way we prove one of the inequalities. Now, for each z 2 ¢ (xs ; ²) let ¿(z) = minfk 2 N : T k (z) 2 ¢ (xs; ²)g. Since ¿ = minz2¢ (xs;²) ¿(z), then ¿(z)¡ 1

¿© (R(xs ; ²; ±)) ¸ ¡ (s + ±) max Á + minz2¢

(x s;²)

¸ ¡ (s + ±) max Á + minz2¢

(x s;²)

X

Á(T j z)

j=0 ¿¡ X1

Á(T j z):

j=0

Furthermore, since Á is a Ho¨lder continuous function, then there exists a constant n n µ > 0 such that jÁ(z) ¡ Á(y)j · e¡ µS (u(¢ (z))) , with n = maxfk 2 N : ¢ k (y) = ¢ k(z)g. Therefore ¿© (R(xs ; ²; ±)) ¸

max

z2¢ (xs ;²)

¿ X j=0

µ

1 Á(T z)¡ (s + ²+ 1) max Á + 1 ¡ exp(¡ µ min Á) j



with max Á = maxz2X Á(z) and similarly for min Á. Finally, the result follows with a = 3 max Á + (1 ¡ exp(¡ µ min Á))¡ 1. 3.3. Proof of Claim 3.1. For each n ¸ 2 let ³ n = f¢ 1; ¢ 2; : : : ; ¢ P (n) g. For each 1 · i · P (n) let n n ±i = e¡ S (u(¢ i)) . Now, for each i 2 f1; 2; : : : ; P(n)g let N(i) = beS (u(¢ i)) (1 ¡ e¡ n max u)¡ 1 c. For i 2 f1; 2; : : : ; P(n)g fixed, and each j 2 f0; 1; : : : ; N(i)g n define s(i; j) = j £ e¡ S (u(¢ i)) (1 ¡ e¡ nmax u), and ²i;j > 0 be such that ys(i;j) 2 S(x(i) ; ² ; ± ) () y 2 ¢ i: Here x(i) 2 ¢ i is taken arbitrary. s(i;j) i;j i

The collection Rn = fR i;j = R(x(i) ; ² ; ± ) : x(i) 2 ¢ i, i 2 f1; 2; : : : ; P(n)g, s(i;j ) i;j i j 2 f0; 1; : : : ; N(i)gg; is a cover of X Á by (squared) rectangles. We use this particular cover to compute an upper bound for the spectra. The bound stated in Remark 3.2 implies that diam Ri;j · maxf½s (xs ; ys ) + jt ¡ sj : xs ; yt 2 Ri;j g:

280

V. Afraimovich, J.-R. Chazottes and E. Ugalde

and taking into account the definition of Ri;j , we obtain M(X Á ; ®; q; R

n) =

PX (n) N(i) X

e¡ q¿Á (Ri;j ) (diam Ri;j )®

i=1 j =0

P (n) M(i)

· e

qa

XX

e

¡ qS ¿i (Á(¢ i ))

i=1 j=0

P (n)

eqa

X

³

e

¡ S n (u(¢ i )))+maxu

+ ±i

´®

N(i)

X

e¡ qS

¿i (Á(¢

i ))

(diam Ri;j )® ;

i=1 j=M(i)+1

with ¿i = ¿T (¢ i) and M(i) = maxf0 · j · N(i) : s(i; M(i)) + ±i · 1g. Now, for j > M(i) and xt ; yt0 2 Ri;j , either jt ¡ t0 j · ±i or y = T (y0 ) for y 0 2 ¢ i . In any case dX Á (xt ; yt0 ) · ±i + dX (T 2 x; T 2y), and from this we have P (n) X ¡ ¢ ¿i n 2 max u ® qa M(X ; ®; q; Rn) · 1 + e e N (i) £ e¡ qS (Á (¢ i ))¡ ® S (u(¢ Á

i ))

:

i=1

n (u(T (¢

Furthermore, since ±j = e¡ S then we have M(X Á ; ®; q; Rn) ·

i ))

and N (i) · eS

n (u(¢

i ))

(1 ¡ e¡ n max u)¡ 1,

P (n) (1 + e2 max u)® qa X (1¡ ® )S n (u(¢ i ))¡ qS¿ i (Á (¢ i )) e e : 1 ¡ e¡ n max u i=1

Now, define Pn;k = f1 · i · P(n) : ¿i = kg. Since (X; T ) is specified, then Pn;k = ; for all k ¸ n + n0. We have, M(X; ®; q; Rn ) ·

n+n (1 + e2 maxu )® qa X0 X (1¡ ® )Sk (u(¢ k (x(i))))¡ qS k (Á (¢ e e 1 ¡ e¡ n maxu

k (x (i) ))

k=1 i2Pn;k

where, as before, x(i) 2 ¢ i is taken arbitrary. For i 6= i0 2 Pn;k there ©necessarily exists 1 ª· j · k such that ¢ j (x(i) ) 6= 0 ¢ j (x(i ) ), which means that ¢ k (x(i) ) : i 2 Pn;k ½ ³ k . Then, taking ± > 0 such that ± + PX((1 ¡ ®)u ¡ qÁjT)) < 0 we obtain 1 (1 + e2 max u)® qa X X (1¡ ® )Sk (u(¢ M(X; ®; q; Rn )· e e 1 ¡ e¡ n max u k k=1

· K± +

+ e2 max u )®

))¡ qSk (Á (¢ ))

¢ 2³ 1 X qa k(±+PX ((1¡ ® )u¡ qÁ jT ))

(1 e 1 ¡ e¡ nmax u

e

k=1

< 1;

;

281

Spectra of Dimensions for Poincar´e Recurrences for Special Flows

for some K± > 0. This bound is independent on n, then M(X Á ; ®; q) · lim M(X Á , n!1 ®, q; Rn) < 1 as far as PX ((1 ¡ ® )u ¡ qÁjT ) < 0. The function ® 7! PX ((1 ¡ ®)u ¡ qÁjT) is non-increasing and the result follows. 3.4. Proof of Claim 3.2. Let R = fR(xs ; ²; ±) : (xs ; ²; ±) 2 BRg be a cover of X Á by rectangles. The elements of this cover are indexed by a finite set BR ½ X Á £ (0; 1)2. Supose that diam R = ²0 for some given ²0 2 (0; 1). m(xs ;²)

(u(¢ (x s;²))) The vertical size. For each (xs ; ²; ±) 2 BR let n(xs ; ²; ±) ´ d±£ eS e, with m(xs ; ²) as in Proposition 3.1 Lemma 3.1 implies that, for each (xs ; ²; ±) 2 BR satisfying 1 ¡ s ¸ ²0 ,

diam R(xs ; ²; ±) ¸ diam S(xs ; ²) + ± ¸ e¡ S

m (u(¢

m (u(¢

For ® ¸ 1 we have (diam R(xs ; ²; ±))® ¸ e¡ ® S M(X Á ; ® ; q; R) ¸

X

(xs ;²)))

(xs;²)))

£ n(xs ; ²; ±):

£ n(xs ; ²; ±); implying

n(xs ; ²; ±)e¡ q¿© (R(xs;²;±))¡ ® S

m(x s ;²) (u(¢

(xs ;²)))

;

(x s;²;±)2B¤R

where B¤R = f(xs ; ²; ±) 2 BR : 1 ¡ s ¸ ²0 g.

According to Lemma 3.2, ¿© (R(xs ; ²; ±)) · a+ S ¿(¢ (xs ;²) (Á(¢ (xs; ²))). Since the (X; T ) is specified, then ¿(¢ (xs ; ²)) · m(xs; ²)+n0. We have, ¿© (R(xs ; ²; ±)) · (a + n0 max Á) + S m(xs;²) Á(¢ (xs ; ²)); and then M(X Á ; ®; q; R) ¸ e¡ q(a+n0 max u)

X

m(xs ;²)((qÁ +® u)(¢

n(xs ; ²; ±)e¡ S

(x s;²)))

:

¤ (xs ;²;±)2BR

On the other hand, since n(xs ; ²; ±) ¸ ±eS M(X Á ; ® ; q; R) ¸ ea+n0 max u

m(xs ;² )(u(¢

X

±eS

(x s;±))) ,

m(x s ;²) ((1¡

then ® )u¡ qÁ )((¢ (xs ;²)))

:

(xs;²;±)2B¤R

The horizontal grouping. Now we will group in the ‘horizontal direction’ ¤ . elements in BR For each N 2 N fixed define the j-th “horizontal slice” [ [ S(N; j) = © (x0; Á(x)t): x2X j=N minf±¡ 1 : (xs ; ²; ±) 2 BRg, #PC(N) ¸ N(1 ¡ ²0) ¡ 2#R. For N > minf±¡ 1 : (xs; ²; ±) 2 BRg, we have X X M(X Á ; ®; q; R) ¸ ea+n0 max u j2P C(N) (x s;²;±)2B(j )

± m(x s ;² )((1¡ ® )u¡ qÁ)(¢ eS CN (xs ; ²; ±)

¸ ¸

ea+n0 maxu N e

a+n0 maxu

N

X

X

(xs ;²)))

m(xs ; ²)((1¡

eS

® )u¡ qÁ)(¢ (xs ;²)))

j2P C(N) (xs;²;±)2B(j )

#P C(N) £

min

C(j) :j2P (N)

¢

X

e¡ S

j¢ j(((1¡

® )u¡ qÁ )(¢ ))

2C(j )

since, as we mentioned before, CN (xs ; ²; ±) · N±. Using the lower bound for #P(N) we obtain µ ¶ X j¢ j #R Á a+n0 maxu M(X ; ®; q; R) ¸ e 1 ¡ ²0 ¡ 2 £ min eS (((1¡ ® )u¡ qÁ )(¢ )) ; C½ CR N ¢ 2C

;

Spectra of Dimensions for Poincar´e Recurrences for Special Flows

283

where the minimum is taken over all the subcovers of the projection CR = f¦ (xs; ²; ±) : (xs ; ²; ±) 2 BRg. Thus, for N ¸ 2#R=²0 , we have X j¢ j M(X Á ; ®; q; R) ¸ ea+n0 max u(1 ¡ 2²0) £ min eS (((1¡ ® )u¡ qÁ )(¢ )) : C½ CR

¢ 2C

As the diameter ²0 goes to zero, the length of the cylinders in CR increase. Indeed, if ¢ 2 CR, then j¢ j ¸ 1 ¡ log(²0 )(min u)¡ 1 . From all this, M(X Á ; ®; q) ¸ C£ ea+n0 max u£ lim (inffZ(0; ((1 ¡ ® )u ¡ qÁ); C; X ) : jCj ¸ ng) ; n!1

with Z as defined in the paragraph 2.7. Hence according to what was exposed in that paragraph, M(X Á ; ®; q) = 1 whenever PX ((1 ¡ ®)u ¡ qÁjT ) > 0. In this way, from Claims 3.1 and 3.2 it follows Theorem 3.2. Now we are in the position to prove our main result, Theorem 3.1. 3.5. Proof of Theorem 3.1. We have that, for all ® ¸ 1 and q ¸ 0, the spectrum for Poincare´ recurrences ® ´ ®(X Á ; q) satisfies the Bowen-like equation PX ((1 ¡ ® ¤ )u ¡ qÁjT ) = 0: ~ e satisfies the On the other hand, the spectrum ® ¹ for the measure ¹ 2 M following equation Z ¡ ¢ h¹ (T) + (1 ¡ ® ¹ )u(x) ¡ qÁ(x) d¹(x) = 0; ¤

X

where ¹ is obtained from an T –ergodic measure ¹ as described before, and h¹ (T) is the entropy of the measure ¹ with respect to T . Now, the classical variational principle for T –invariant measures (see for instance [12]), applied to the potential (1 ¡ ® )u ¡ qÁ, establishes that ½ ¾ Z PX ((1 ¡ ®)u ¡ qÁjT) = sup h¹ (T) + ((1 ¡ ®)u(x) ¡ qÁ(x)) d¹(x) ; ¹ 2M(T )

X

where M(T ) is the set of all T-invariant measures. Now, for q ¸ 0 fixed and taking ® = ® ¤ := ® c(X Á ; q) in both sides of the previous equation, we have ½ Z ¾ 0 = PX((1¡ ® ¤ )u¡ qÁjT) = sup hº (T ) + ((1 ¡ ® ¤ )u(x) ¡ qÁ(x)) d¹(x) ; R

¹ 2M

X

which implies that hº (T )+ X ((1¡ ® ¤ )u(x)¡ qÁ(x)) d¹(x) · 0R for all ¹ 2 M(T ). Since, for each º fixed and u > 0, the function ® ! hº (T) + X ((1 ¡ ® ¤ )u(x) ¡ qÁ(x)) d¹(x) is decreasing, then necessarily ® ¤ ¸ supf® ¹ (q) : ¹ 2 Me g:

284

V. Afraimovich, J.-R. Chazottes and E. Ugalde

On the other hand, if ® < ® ¤ then PX((1 ¡ ® ¤ )u ¡ qÁjT) > 0, implying that ½ Z ¾ sup hº (T) + ((1 ¡ ®)u(x) ¡ qÁ(x)) dº(x) > 0: º 2M

X

R Thus, there necessarily exists a measure º 2 M(T) such that hº (T ) + X ((1 ¡ ®)u(x) ¡ qÁ(x))dº(x) > 0. Again, for º 2 M(T) fixed and u > 0, the function R ® ! hº (T ) + X ((1 ¡ ®)u(x) ¡ qÁ(x))dº(x) is decreasing, then ® < ® º¹ (q). In this way we obtain the reverse inequality, i. e., ® ¤ · supf® ¹ (q) : ¹ 2 Meg; and the theorem follows. ACKNOWLEDGMENTS V. A. was partially supported by the NSF–CONACyT grant no. E120.0547 and 2001 UC MEXUS–CONACyT grant. E. U. was supported by CONACyT grant no. J32389E. The authors thank H. Weiss and J. Schmeling for the possibility to read the manuscript [11]. REFERENCES 1. V. Afraimovich, Pesin’s dimension for Poincare´ recurrences, Chaos 7, (1997) 12-20. 2. V. Afraimovich, J. Schmeling, E. Ugalde and J. Urí as, Spectra of dimensions for Poincar ´e recurrences, Discrete and Continuous Dynamical Systems 6 no.4, (2000), 901-914. 3. L. Barreira and J. Schmeling, Sets of “non-typical”points have full topological entropy and full Hausdorff dimension, Israel J. Math. 116 (2000), 29-70. 4. V. Afraimovich, J.-R. Chazottes and B. Saussol, Local dimensions for Poincar e´ recurrences, Electronic Research Announcements of the AMS 6 (2000), 64-74. 5. R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429-460. 6. R. Bowen and P. Walters, Expansive one-parameter flows, J. Diff. Equas. 12 (1972), 180-193. 7. J.-R. Chazottes, Poincar ´e recurrences and the entropy of suspended flows, C. R. Acad. Sci. Paris Se´r. I Math. 332 (2001), no. 8, 739-744. 8. W. Parry, M. Pollicott, “Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics”, Aste´risque 187-188, SMF, 1990. 9. Ya. B. Pesin, Dimension theory in dynamical systems, contemporary views and applications, Chicago Lectures in Mathematics (1997). 10. Ya. Pesin and V. Sadovskaya, Multifractal analysis of conformal axiom A flows, Comm. Math. Phys. 216 no. 2 (2001), 277-312.

Spectra of Dimensions for Poincar´e Recurrences for Special Flows

285

11. J. Schmeling and H. Weiss. Minicourse in dynamical systems and dimensions theory, to appear. 12. P. Walters, An introduction to ergodic theory, Springer-Verlag (1982). 13. L.-S. Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153-188.

V. Afraimovich, J.-R. Chazottes and E. Ugalde IICO, Universidad Auto´noma de San Luis Potosí 78000 San Luis Potosí, SLP, Mexico (V. A. and E. U.) and ´ CPHT, Ecole Polytechnique, 91128 Palaiseau Cedex, France (J.-R. C)

Lihat lebih banyak...

Comentários

Copyright © 2017 DADOSPDF Inc.