Abstract
In this paper we unify and extend many of the known results on the dimension of deterministic and random Cantor-like sets in ℝn, and apply these results to study some problems in dynamical systems. In particular, we verify the Eckmann-Ruelle Conjecture for equilibrium measures for Hölder continuous conformal expanding maps and conformal Axiom A# (topologically hyperbolic) homeomorphims. We also construct a Hölder continuous Axiom A# homeomorphism of positive topological entropy for which the unique measure of maximal entropy is ergodic and has different upper and lower pointwise dimensions almost everywhere. this example shows that the non-conformal Hölder continuous version of the Eckmann-Ruelle Conjecture is false.
The Cantor-like sets we consider are defined by geometric constructions of different types. The vast majority of geometric constructions studied in the literature are generated by a finite collection ofp maps which are either contractions or similarities and are modeled by the full shift onp symbols (or at most a subshift of finite type). In this paper we consider much more general classes of geometric constructions: the placement of the basic sets at each step of the construction can be arbitrary, and they need not be disjoint. Moreover, our constructions are modeled by arbitrary symbolic dynamical systems. The importance of this is to reveal the close and nontrivial relations between the statistical mechanics (and especially the absence of phase transitions) of the symbolic dynamical system underlying the geometric construction and the dimension of its limit set. This has not been previously observed since no phase transitions can occur for subshifts of finite type.
We also consider nonstationary constructions, random constructions (determined by an arbitrary ergodic stationary distribution), and combinations of the above.
Similar content being viewed by others
References
[AJ] Alekseyev, V., Jakobson, M.: Symbolic dynamics and Hyperbolic Dynamical Systems. Phys. Rep.75, 287–325 (1981)
[AS] Afraimovich, V., Shereshevsky, M.: The Hausdorff Dimension of Attractors Appearing by Saddle-Node Bifurcations. Int. J. Bifurcation and Chaos1:2, 309–325 (1991)
[Ba] Barreira, L.: Cantor Sets with Complicated Geometry and Modeled by General Symbolic Dynamics. To appear, Random and Computational Dynamics (1995)
[BFKO] Bourgain, J.: Pointwise Ergodic Theorems for Arithmetic Sets, with an Appendix by J. Bourgain, H. Furstenberg, Y. Katznelson, and D. Ornstein. Publ. Math. IHES69, 5–45 (1989)
[BK] Brin, M., Katok, A.: On Local Entropy. Lecture Notes in Mathematics1007, Geometric Dynamics (1981), Berlin-Heidelberg-New York, Springer Verlag, pp. 30–38
[Bo1] Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Springer Lecture Notes #470, Berlin-Heidelberg-New York, Springer Verlag, 1975
[Bo2] BowBo, R.: Hausdorff Dimension of Quasi-circles. Publ. Math. IHES50, 11–25 (1979)
[BM] Bertrand-Mathis, A.: Questions Diverses Relatives aux Systèms codés: Applications au δ-shift. Preprint
[BU] Bedford, T., Urbański, M.: The box and Hausdorff Dimension of Self-Affine Sets. Ergod. Th. and Dynam. Systems10, 627–644 (1990)
[C] Cutler, C.D.: Connecting Ergodicity and Dimension in Dynamical Systems. Ergod. Th. and Dynam. Systems10, 451–462 (1990)
[CLP] Collet, P., Lebowitz, J.L., Porzio, A.: The Dimension Spectrum of Some Dynamical Systems. J. Stat. Physics47, 609–644 (1987)
[CM] Cawley, R., Mauldin, R.: Adv. Math.92, 196–236 (1992)
[EM] Mauldin, R., Edgar, G.: Multifractal Decomposition of Digraph Recursive fractals. Proc. London Math. Soc.65, 604–628 (1992)
[ER] Eckmann, J.P., Ruelle, D.: Ergodic Theory of Chaos and Strange Attractors. 3. Rev. Mod. Phys.57, 617–656 (1985)
[F1] Falconer, K.: Fractal Geometry, Mathematical Foundations and Applications Cambridge: Cambridge Univ. Press, 1990
[F2] Falconer, K.: Random Fractals. Math. Proc. Camb. Phil. Soc.100, 559–582 (1986)
[Fe] Federer, H.: Geometric Measure Theory. Berlin-Heidelberg-New York: Springer Verlag, 1969
[Fr] Frostman, O.: Potential d'équilibre et Capacité des Ensembles Avec Quelques Applications à la Théorie des Fonctions. Meddel. Lunds Univ. Math. Sem.3, 1–118 (1935)
[Fu] Furstenberg, H.: Disjointness in Ergodic Theory, Minimal Sets, and a Problem in Diophantine Approximation. Mathematical Systems Theory1, 1–49 (1967)
[G] Graf, S.: Statistically Self-similar Fractals. Prob. Theory and Related Fields74, 357–397 (1987)
[GMW] Graf, S., Mauldin, D., Williams, S.: The Exact Hausdorff Dimension in Random Recursive Constructions. 381, Mem. Am. Math. Soc.71 (1988)
[KH] Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press, 1995
[HP] Hentschel, H.G.E., Procaccia, I.: The Infinite Number of Generalized Dimensions of Fractals and Strange Attractors. Physica8D, 435–444 (1983)
[K] Kahane, J.P.: Sur le Modéle de Turbulence de Benoit Mandelbrot. C. R. Acad. Sci. Paris278A, 621–623 (1974)
[L] Ledrappier F.: Dimension of Invariant Measures. Preprint (1992)
[Lo] Lopes, A.: The Dimension Spectrum of the Maximal Measure. SIAM J. Math. Analysis20, 1243–1254 (1989)
[LM] Ledrappier, F., Misiurewicz, M.: Dimension of Invariant Measures for Maps with Exponent Zero. Ergod. Th. and Dynam. Systems5, 595–610 (1985)
[LY] Ledrappier, F., Young, L.S.: The Metric Entropy of Diffeomorphisms. Part II. Ann. of Math.122, 540–574 (1985)
[Mc] McMullen, C.: The Hausdorff Dimension of General Sierpiński Carpets. Nagoya Math. J.96, 1–9 (1984)
[MW1] Mauldin, R., Williams, S.: Hausdorff Dimension in Graph Directed Constructions. Trans. AMS309:2, 811–829 (1988)
[MW2] Mauldin, R., Williams, S.: Random Geometric Constructions: Asymptotic, Geometric and Asymptotic Properties. Trans. Am. Math. Soc.295, 325–346 (1986)
[Mo] Moran, P.: Additive Functions of Intervals and Hausdorff Dimension. Proc. Camb. Phil. Society42, 15–23 (1946)
[OSY] Ott, E., Sauer, T., Yorke, J.: Lyapunov Partition Functions for the Dimensions of Chaotic Sets. no 9, Phys. Rev.A39, 4212–4222 (1989)
[PP] Parry, W., Pollicott, W.: Zeta Functions and the Periodic, Orbit Structures of Hyperbolic Dynamics. Astérisque187–188, (1990)
[PW1] Pesin, Y., Weiss, H.: On the Dimension of Deterministic and Random Cantor-like Sets. Math. Res. Lett.1, 519–529 (1994)
[PW2] Pesin, Y., Weiss, H.: A Multifractal Analysis of Equilibrium Measures for Conformal Expanding Maps and Markov Moran Geometric Constructions Preprint (1995)
[PoW] Pollicott, M., Weiss, H.: The Dimensions of Some Self-affine Limit Sets in the Plane and Hyperbolic Sets. Preprint (1993)
[PU] Przytycki, F., Urbański, M.: On Hausdorff Dimension of Some Fractal Sets. Studia Mathematica93, 155–186 (1989)
[PY] Pesin, Y., Yue, C.B.: Hausdorff Dimension of Measures with Non-zero Lyapunov Exponents and Local Product Structure. PSU Preprint (1993)
[R] Ruelle, D.: Thermodynamic Formalism. Reading, Addison-Wesley, 1978
[S] Shereshevsky, M.: On the Hausdorff Dimension of a Class of Non-Self-Similar Fractals. Math. Notes50, no. 5, 1184–1187 1991
[Sh] Shub, M.: Global Stability of Dynamical Systems. Berline-Heidelberg-New York: Springer Verlag, 1987
[St] Stella, S.: On Hausdorff Dimension of Recurrent Net Fractals. Proc. Am. Math. Soc.116, 389–400 (1992)
[Y] Young, L.S.: Dimension, Entropy, and Lyapunov Exponents. Ergod. Th. and Dynam. Systems2, 109–124 (1982)
[W] Walters, P.: Introduction to Ergodic Theory. Berlin-Heidelberg-New York: Springer Verlag, 1982
[We] Weiss, B.: Subshifts of Finite Type and Sophic Systems. Monatschefte für Mathematik77, 462–474 (1973)
[W1] Weiss, H.: The Multifractal Analysis of Topologically Hyperbolic Homeomorphisms. In preparation
Author information
Authors and Affiliations
Additional information
Communicated by Ya.G. Sinai
The work of the first author was partially supported by a National Science Foundation grant #DMS91-02887. The work of the second author was partially supported by a National Science Foundation Postdoctoral Research Fellowship and National Science Foundation grant #DMS-9403724.
Rights and permissions
About this article
Cite this article
Pesin, Y., Weiss, H. On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle Conjecture. Commun.Math. Phys. 182, 105–153 (1996). https://doi.org/10.1007/BF02506387
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02506387