Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann--Ruelle conjecture, Comm. Math. Phys. 182 (1996), 105--153.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....x such that #(y, r) #(x) for every y F , and #(x) #(z, r) for some z F . The sets #(x) form a cover of F that we call the (#, r) Moran cover of F . We note that under the assumption (14) the sets #(x) are pairwise disjoint. Related covers were introduced by Pesin and Weiss in [14] (see [13] for details) Moran covers are used in the remaining sections of the paper. The name is due to the seminal work of Moran in [12] where he considered geometric constructions modelled by Q = #, such that each set # #n is a ball of radius k=1 # i k for some fixed numbers # 1 , # ....

....#n . This is in strong contrast to what happens for generalized Moran constructions. See [1] for other examples of weakly regular constructions and for criteria of weak regularity. We note that the class of weakly regular contains the class of regular constructions introduced by Pesin and Weiss in [14]. 6.2. Lower dimension estimates. We now obtain a lower dimension estimate for the limit set of a weakly regular geometric construction. Consider a sequence of positive number # = ##Q,n#N as in Section 6.1. Define a sequence # of functions # n : Q R by # n (#) log # #n . Let s P be the ....

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann--Ruelle conjecture, Comm. Math. Phys. 182 (1996), 105--153.

....x such that #(y, r) #(x) for every y F , and #(x) #(z, r) for some z F . The sets #(x) form a cover of F that we call the (#, r) Moran cover of F . We note that under the assumption (14) the sets #(x) are pairwise disjoint. Related covers were introduced by Pesin and Weiss in [13] (see [12] for details) Moran covers are used in the remaining sections of the paper. The name is due to the seminal work of Moran in [11] where he considered geometric constructions modelled by Q = #, such that each set # #n is a ball of radius k=1 # i k for some fixed numbers # 1 , # ....

....#n . This is in strong contrast to what happens for generalized Moran constructions. See [1] for other examples of weakly regular constructions and for criteria of weak regularity. We note that the class of weakly regular contains the class of regular constructions introduced by Pesin and Weiss in [13]. 6.2. Lower dimension estimates. We now obtain a lower dimension estimate for the limit set of a weakly regular geometric construction. Consider a sequence of positive number # = ##Q,n#N as in Section 6.1. Define a sequence # of functions # n : Q R by # n (#) log # #n . Let s P be the ....

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann--Ruelle conjecture, Comm. Math. Phys. 182 (1996), 105--153.

....Furthermore, the hyperbolic measure may not possess a uniform product structure even if the support does. We know that the hypotheses in Theorem 5 are in a certain sense optimal: Ledrappier and Misiurewicz [62] showed that the hyperbolicity of the measure is essential, while Pesin and Weiss [85] showed that the statement in Theorem 5 cannot be extended to Holder homeomorphisms. On the other hand one does not know what happens for C di#eomorphisms that are not of class C for some # 0, particularly due to the nonexistence of an appropriate theory of nonuniformly hyperbolic ....

....i k . We define the set In [70] Moran showed that dimH F = s where s is the unique real number satisfying the identity = 1. 12) It is remarkable that the Hausdor# dimension of F does not depend on the location of the intervals # i 1 n but only on their length. Pesin and Weiss [85] extended the result of Moran to arbitrary symbolic dynamics in R using the thermodynamic formalism (see Section 4.2) To model hyperbolic invariant sets, we need to consider geometric constructions described in terms of symbolic dynamics. Given an integer p 0, we consider the family of ....

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Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann--Ruelle conjecture, Comm. Math. Phys. 182 (1996), 105--153.

....dimH d (x) for almost every x 2 X. Therefore, Theorem 3 may in general provide a stronger statement than that in (3) and the rst inequality in (2) may 4 L. BARREIRA AND B. SAUSSOL be sharper than that in Theorem 2. This possibility indeed occurs in the following example. Example 1. In [10], Pesin and Weiss presented an example of a H older homeomorphism in a closed subset X of [0; 1] whose unique (and thus ergodic) measure of maximal entropy is not exact dimensional. More precisely, there exist disjoint sets A 1 , A 2 [0; 1] with positive measure whose union is equal to X, ....

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann{Ruelle conjecture, Comm. Math. Phys. 182 (1996), 105-153.

....i.e. d = d = d s d u : Let us also point out that neither of the assumptions of the Main Theorem can be omitted. Ledrappier and Misiurewicz [24] constructed an example of a smooth map of a circle preserving an ergodic measure with zero Lyapunov exponent which is not exact dimensional. In [33], Pesin and Weiss presented an example of a H older homeomorphism with H older constant arbitrarily close to 1 whose measure of maximal entropy is not exact dimensional. Statement 1 of the Main Theorem establishes a new and non trivial property of an arbitrary hyperbolic measure. Loosely speaking ....

Ya. Pesin, and H. Weiss, On the dimension of deterministic and random Cantorlike sets, symbolic dynamics, and the Eckmann{Ruelle conjecture, Comm. Math. Phys., vol 182 (1996), 105-153

....[13] one can show that dimH d (x) for almost every x 2 X . Therefore, Theorem 3 may in general provide a stronger statement than that in (5) and the rst inequality in (3) may be sharper than that in Theorem 2. This possibility indeed occurs in the following example. Example 1. In [10], Pesin and Weiss presented an example of a H older homeomorphism in a closed subset X of [0; 1] whose unique (and thus ergodic) measure of maximal entropy is not exact dimensional. More precisely, there exist disjoint sets A 1 , A 2 [0; 1] with positive measure whose union is equal to X , ....

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann{Ruelle conjecture, Comm. Math. Phys. 182 (1996), 105-153.

....and let Q p be a compact shift invariant subset. We de ne a set F = F (Q) in a way similar to that in (1) where the union is taken only over the Q admissible tuples (i 1 i n ) that is, those tuples such that (i 1 i n ) j 1 j n ) for some (j 1 j 2 ) 2 Q. In [PeW], Pesin and Weiss showed that if the intervals i 1 i n have length Q n k=1 i k , then the identities in (3) hold when we replace s by the unique root of Bowen s equation P (s ) 0: 4) Here is the function de ned on Q by (i 1 i 2 ) log i 1 , and P is the topological ....

....r i 1 i n and positive constants C 1 C 2 . We notice that the numbers r i 1 i n may depend on all the symbolic past, and may satisfy no asymptotic behavior. A special class of generalized Moran constructions is the class of Moran constructions, introduced in [M] when Q = p (see [PeW]) We de ne the sequence of functions n (i 1 i 2 ) log r i 1 i n on Q, and denote it . Then property (6) holds for the shift map. We assume that there exist K 1 and 0 such that, for each (i 1 i 2 ) 2 Q and n 1, r i 1 i n K n and r i 1 i n 1 r i 1 i n : ....

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture, Comm. Math. Phys. (to appear).

....1 i n . The next step is to consider intervals i 1 i n with length r i 1 i n that may satisfy no asymptotic behavior, and may depend on all the symbolic past. This situation has never been studied before, except for very special constructions exhibiting a strong asymptotic behavior (see [PeW1]) We de ne the limit set F as in (1) Its Hausdor dimension dimH F may not coincide with its lower and upper box dimensions dim B F and dimB F . Let (i 1 i 2 ) i 2 i 3 ) be the shift map. If the sequence of functions n (i 1 i 2 ) log r i 1 i n is sub additive, that ....

....and (j 1 j 2 ) in p is P n 1 2 n ji n j n j) We de ne a set (Q) F in a way similar to that in (1) where the union is taken only over the Q admissible tuples (i 1 i n ) that is, those tuples such that (i 1 i n ) j 1 j n ) for some (j 1 j 2 ) 2 Q. In [PeW1], Pesin and Weiss showed that if the intervals i 1 i n have length Q n k=1 i k , then the identities in (3) hold when we replace F by (Q) and s by the unique root of Bowen s equation P (s ) 0: 5) Here is the function de ned on Q by (i 1 i 2 ) log i 1 and P is the ....

[Article contains additional citation context not shown here]

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture, Comm. Math. Phys. (to appear).

....can be reduced to one constructed with aligned rectangles, having equal Hausdor and box dimensions. This is proved showing the existence of a vector eld along which the rectangles are forced to orient. INTRODUCTION A new line of research has recently emerged from the work of Pesin and Weiss [5], where they study the Hausdor dimension, and lower and upper box dimensions of a class of Cantor sets. These are constructed with a decreasing sequence of closed sets of which we only know some geometric properties, such as the radii of inscribed and circumscribed circles of their connected ....

....Set Q = A and, for each (i 1 i 2 ) 2 Q and n 2 N, de ne basic sets i 1 i n = T n j=1 f j 1 i j . The limit set is the f invariant set T 1 n=1 f n and is called dynamically de ned Cantor set. For more examples as well as the description of some pathologies see [5] and [1] If 2 Q, we write = i 1 ( i 2 ( or simply = i 1 i 2 ) when there is no danger of confusion. Put ( 1 n=1 i 1 i n : 2) By Condition (b) j i 1 i n j 0 as n 1 and hence, by (1) is a point in F . It follows that (2) gives a well de ....

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Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantorlike sets, symbolic dynamics, and the Eckmann-Ruelle conjecture, Preprint, PSU (1994).

....We define a set F = F (Q) in a way similar to that in (1) where the union is taken only over the Q admissible tuples (i 1 Delta Delta Delta i n ) that is, those tuples such that (i 1 Delta Delta Delta i n ) j 1 Delta Delta Delta j n ) for some (j 1 j 2 Delta Delta Delta ) 2 Q. In [PeW], Pesin and Weiss showed that if the intervals Delta i 1 Delta Delta Deltai n have length Q n k=1 i k , then the identities in (3) hold when we replace s by the unique root of Bowen s equation P (s ) 0: 4) Here is the function defined on Q by (i 1 i 2 Delta Delta Delta ) log i ....

....positive constants C 1 C 2 . We notice that the numbers r i 1 Delta Delta Deltai n may depend on all the symbolic past, and may satisfy no asymptotic behavior. A special class of generalized Moran constructions is the class of Moran constructions, introduced in [M] when Q = Sigma p (see [PeW]) We define the sequence of functions n (i 1 i 2 Delta Delta Delta ) log r i 1 Delta Delta Deltai n on Q, and denote it Phi. Then property (6) holds for the shift map. We assume that there exist K 1 and ffi 0 such that, for each (i 1 i 2 Delta Delta Delta ) 2 Q and n 1, r i 1 ....

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture, Comm. Math. Phys. (to appear).

....can be reduced to one constructed with aligned rectangles, having equal Hausdorff and box dimensions. This is proved showing the existence of a vector field along which the rectangles are forced to orient. INTRODUCTION A new line of research has recently emerged from the work of Pesin and Weiss [5], where they study the Hausdorff dimension, and lower and upper box dimensions of a class of Cantor sets. These are constructed with a decreasing sequence of closed sets of which we only know some geometric properties, such as the radii of inscribed and circumscribed circles of 1991 Mathematics ....

.... Delta ) 2 Q and n 2 N, define basic sets Delta i 1 Delta Delta Deltai n = T n j=1 f Gammaj 1 Delta i j . The limit set is the f invariant set T 1 n=1 f Gamman Delta and is called dynamically defined Cantor set. For more examples as well as the description of some pathologies see [5] and [1] If 2 Q, we write = Gamma i 1 ( i 2 ( Delta Delta Delta Delta or simply = i 1 i 2 Delta Delta Delta ) when there is no danger of confusion. Put ( 1 n=1 Delta i 1 Delta Delta Deltai n : 2) By Condition (b) j Delta i 1 Delta Delta Deltai n j 0 as n ....

[Article contains additional citation context not shown here]

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture, Preprint, PSU (1994).

No context found.

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann--Ruelle conjecture, Comm. Math. Phys. 182 (1996), 105--153.

No context found.

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann--Ruelle conjecture, Comm. Math. Phys. (to appear).

....by a National Science Foundation Postdoctoral Research Fellowship. Typeset by AM S T E X 0: Introduction. In this paper we unify and extend many of the known results on the Hausdorff and box dimension of deterministic and random Cantor like sets in R determined by geometric constructions (see [PW] for the complete description of results and detailed proofs) Most authors have considered similarity processes which impose a strong restriction on the geometry of the construction. Moreover, these constructions were modeled by either the full shift, or subshifts of finite type. In this paper we ....

....and spacing of the basic sets. We control the geometry of the construction by either restricting the shapes or sizes of the basic sets, the spacing of the basics sets, or both. If one has strong control over the sizes of the basic sets, then the spacing can be fairly arbitrary, and vice versa. In [PW], we introduce a new approach to studying geometric constructions having complicated geometry. Our approach is based on the notions of regularity and boundedness of the construction. Regular and bounded constructions are those where the control over the geometry is effected in the spirit of Moran ....

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Y. Pesin and H. Weiss, On the Dimension Of Deterministic and Random Cantor-like sets, Symbolic Dynamics, and the Eckmann-Ruelle Conjecture, Submitted for publication (1994).

....#) is the smallest number of balls of radius # needed to cover the set Z. It is easy to see that dimH Z # dim B Z # dimB Z. The coincidence of the Hausdor# dimension and lower and upper box dimension is a relatively rare phenomenon and can occur only in some rigid situations (see [1] 5] [19]) In order to describe the geometric structure of a subset Z invariant under a dynamical system f acting on X we consider a measure supported on Z. Its Hausdor# dimension and lower and upper box dimensions (which are denoted by dimH , dim B , and dimB , respectively) are dimH = inf dim H Z : ....

....1 holds in this case as well. Let us also point out that neither of the assumptions of the Main Theorem can be omitted. Ledrappier and Misiurewicz [15] constructed an example of a smooth map of a circle preserving an ergodic measure with zero Lyapunov exponent which is not exact dimensional. In [19], Pesin and Weiss presented an example of a Holder homeomorphism with Holder constant arbitrarily close to 1 whose measure of maximal entropy is not exact dimensional. Remarks. 1. Statement 1 of the Main Theorem establishes a new and nontrivial property of an arbitrary hyperbolic measure. Loosely ....

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture, Comm. Math. Phys. 182 (1996), 105--153.

.... difference between the two notions of dimension is that while the definition of pointwise dimension uses arbitrary intervals, the Markov pointwise dimension uses only those intervals of the form I n (x) For arbitrary invariant measures, there are subtleties in trying to relate the quantities [PW1, PW3], but Proposition 3 says that for equilibrium states, the two quantities, as long as they are well defined, coincide. Proof of Proposition 3. The proof of (1) follows immediately from the Jacobian Estimate (J) and (2) The proof of (2) is modeled on the proof of Theorem 7 in [PW3] The Jacobian ....

....quantities [PW1, PW3] but Proposition 3 says that for equilibrium states, the two quantities, as long as they are well defined, coincide. Proof of Proposition 3. The proof of (1) follows immediately from the Jacobian Estimate (J) and (2) The proof of (2) is modeled on the proof of Theorem 7 in [PW3]. The Jacobian estimate (J) allows us to estimate the lengths ( of the intervals I n (x) using the derivative of T [PW1] i.e. there exist positive constants C 1 and C 2 such that for all x 2 I n O and all n 2 N , we have that C 1 (I n (x) j(T n ) 0 (x)j Gamma1 C 2 : 4) Suppose ....

Y. Pesin and H. Weiss, On the Dimension of Deterministic and Random Cantor-like sets, Symbolic Dynamics, and the Eckmann-Ruelle Conjecture, Comm. Math. Physics 182 (1996), 105--153..

....where N(Z; is the smallest number of balls of radius needed to cover the set Z. It is easy to see that dimH Z dim B Z dimB Z: The coincidence of the Hausdor dimension and lower and upper box dimension is a relatively rare phenomenon and can occur only in some rigid situations (see [1, 5, 19]) In order to describe the geometric structure of a subset Z invariant under a dynamical system f acting on X we consider a measure supported on Z. Its 4 LUIS BARREIRA, YAKOV PESIN, AND J ORG SCHMELING Hausdor dimension and lower and upper box dimensions (which are denoted by dimH , dim ....

....1 holds in this case as well. Let us also point out that neither of the assumptions of the Main Theorem can be omitted. Ledrappier and Misiurewicz [15] constructed an example of a smooth map of a circle preserving an ergodic measure with zero Lyapunov exponent which is not exact dimensional. In [19], Pesin and Weiss presented an example of a H older homeomorphism with H older constant arbitrarily close to 1 whose measure of maximal entropy is not exact dimensional. Remarks. 1) Statement 1 of the Main Theorem establishes a new and non trivial property of an arbitrary hyperbolic measure. ....

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann{Ruelle conjecture, Comm. Math. Phys. 182 (1996), 105-153.

....stable and unstable measures. Remarks. 1. Let us point out that neither of the assumptions of the theorem can be omitted. Ledrappier and Misiurewicz [LM] constructed an example of a smooth map of a circle preserving an ergodic measure with zero Lyapunov exponent which is not exact dimensional. In [PW], Pesin and Weiss presented an example of a Holder homeomorphism whose measure of maximal entropy is not exact dimensional. 2. It follows immediately from the theorem that the pointwise dimension of an ergodic invariant measure supported on a (uniformly) hyperbolic locally maximal set is exact ....

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann--Ruelle conjecture, Comm. Math. Phys. (to appear).

....a non negative constant. If is exact dimensional then Young [Y] showed that the Hausdorff dimension of coincides with the box dimension of . In general one does not expect the pointwise dimension of to exist at a typical point even for nice measures which are invariant under dynamical systems [LM, PW]. Even when the pointwise dimension of does exist it is not necessarily exact dimensional [C, PW] Nevertheless, measures which are invariant under smooth dynamical systems with hyperbolic behavior often turn out to be exact dimensional. Eckmann and Ruelle have conjectured that hyperbolic ....

....of coincides with the box dimension of . In general one does not expect the pointwise dimension of to exist at a typical point even for nice measures which are invariant under dynamical systems [LM, PW] Even when the pointwise dimension of does exist it is not necessarily exact dimensional [C, PW]. Nevertheless, measures which are invariant under smooth dynamical systems with hyperbolic behavior often turn out to be exact dimensional. Eckmann and Ruelle have conjectured that hyperbolic measures (i.e. ergodic measures with non zero Lyapunov exponents almost everywhere) are exact ....

[Article contains additional citation context not shown here]

Y. Pesin and H. Weiss, On the Dimension of Deterministic and Random Cantor-like Sets, Math. Research Letters 1 (1994), 519--529.

....and denote it by d (x) We call exact dimensional if d (x) d (x) d (x) c for Gammaalmost every x where c is a non negative constant. For a general dynamical system one does not expect the pointwise dimension of an invariant measure to exist at a typical point, even for nice measures [LM, PW2, PW3]. Even when the pointwise dimension of does exist need not be exact dimensional [C, PW2] Nevertheless, measures which are invariant under smooth dynamical systems with hyperbolic behavior often turn out to be exact dimensional. Eckmann and Ruelle have conjectured that hyperbolic measures (i.e. ....

Y. Pesin and H. Weiss, On the Dimension of Deterministic and Random Cantor-like Sets, Symbolic Dynamics, and the Eckmann-Ruelle Conjecture, Submitted for publication (1994).

....and denote it by d (x) We call exact dimensional if d (x) d (x) d (x) c for Gammaalmost every x where c is a non negative constant. For a general dynamical system one does not expect the pointwise dimension of an invariant measure to exist at a typical point, even for nice measures [LM, PW2, PW3]. Even when the pointwise dimension of does exist need not be exact dimensional [C, PW2] Nevertheless, measures which are invariant under smooth dynamical systems with hyperbolic behavior often turn out to be exact dimensional. Eckmann and Ruelle have conjectured that hyperbolic measures (i.e. ....

....every x where c is a non negative constant. For a general dynamical system one does not expect the pointwise dimension of an invariant measure to exist at a typical point, even for nice measures [LM, PW2, PW3] Even when the pointwise dimension of does exist need not be exact dimensional [C, PW2]. Nevertheless, measures which are invariant under smooth dynamical systems with hyperbolic behavior often turn out to be exact dimensional. Eckmann and Ruelle have conjectured that hyperbolic measures (i.e. ergodic measures with non zero Lyapunov exponents almost everywhere) are exact ....

Y. Pesin and H. Weiss, On the Dimension of Deterministic and Random Cantor-like Sets, Math. Research Letters 1 (1994), 519--529.

No context found.

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture, Comm. Math. Phys. (to appear).

No context found.

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann--Ruelle conjecture, Comm. Math. Phys. 182 (1996), 105--153.

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