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A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems

Published online by Cambridge University Press:  14 October 2010

Luis M. Barreira
Luis M. Barreira
Affiliation:
Department of Mathematics, The Pennsylvania State UniversityUniversity Park, PA 16802USA (e-mail: [email protected])
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Abstract

A non-additive version of the thermodynamic formalism is developed. This allows us to obtain lower and upper bounds for the dimension of a broad class of Cantor-like sets. These are constructed with a decreasing sequence of closed sets that may satisfy no asymptotic behavior. Moreover, they can be coded by arbitrary symbolic dynamics, and the geometry of the construction may depend on all the symbolic past. Applications include estimates of dimension for hyperbolic sets of maps that need not be differentiable.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 5 , October 1996 , pp. 871 - 927
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[AY]Alekseev, V. and Yakobson, M.. Symbolic dynamics and hyperbolic dynamic systems. Phys. Rep. 75 (1981), 287–325.CrossRefGoogle Scholar
[Ba]Barreira, L.. Dimension of Cantor sets with complicated geometry Equadiff 95 Proceedings. To appear.Google Scholar
[Bol]Bowen, R.. Topological entropy for noncompact sets. Trans. Amer. Math. Soc. 184 (1973), 125–136.CrossRefGoogle Scholar
[Bo2]Bowen, R.. Equilibrium Slates and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics 470). Springer, 1975.CrossRefGoogle Scholar
[Bo3]Bowen, R.. Hausdorff dimension of quasi-circles. Inst. Hautes Études Sci. Publ. Math. 50 (1979), 259–273.CrossRefGoogle Scholar
[Fl]Falconer, K.. A subadditive thermodynamic formalism for mixing repellers. J. Phys. A: Math. Gen. 21 (1988), L737–L742.CrossRefGoogle Scholar
[F2]Falconer, K.. Dimensions and measures of quasi self-similar sets. Proc. Amer. Math. Soc. 106 (1989), 543–554.CrossRefGoogle Scholar
[F3]Falconer, K.. Bounded distortion and dimension for non-conformal repellers. Math. Proc. Cambridge Philos. Soc. 115 (1994), 315–334.CrossRefGoogle Scholar
[GP]Gatzouras, D. and Peres, Y.. Invariant measures of full dimension for some expanding maps. Preprint 1994.Google Scholar
[HP]Hirsch, M. and Pugh, C.. Stable manifolds and hyperbolic sets, Global Analysis (Proceeding of Symposia in Pure Mathematics XIV). American Mathematical Society, 1970, pp. 133–163.CrossRefGoogle Scholar
[KH]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications 54). Cambridge University Press, 1995.Google Scholar
[K]Kifer, Y.. Characteristic exponents of dynamical systems in metric spaces. Ergod. Th. & Dynam. Sys. 3 (1983), 119–127.CrossRefGoogle Scholar
[Ma]Mañé, R.. Ergodic Theory and Differentiable Dynamics (Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 8). Springer, 1987.CrossRefGoogle Scholar
[MM]McCluskey, H. and Manning, A.. Hausdorff dimension for horseshoes. Ergod. Th. & Dynam. Sys. 3 (1983), 251–260.CrossRefGoogle Scholar
[Mo]Moran, P.. Additive functions of intervals and Hausdorff measure. Proc. Cambridge Philos. Soc. 42 (1946), 15–23.CrossRefGoogle Scholar
[PT]Palis, J. and Takens, F.. Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, 1993.Google Scholar
[PV]Palis, J. and Viana, M.. On the continuity of Hausdorff dimension and limit capacity for horseshoes. Dynamical Systems (Valparaiso 1986), (Lecture Notes in Mathematics 1331). Eds Bamón, R., Labarca, R. and Palis, J.Jr. Springer, 1988, pp. 150–160.CrossRefGoogle Scholar
[Pa]Parry, W.. Symbolic dynamics and transformations of the unit interval. Trans. Amer. Math. Soc. 122 (1966), 368–378.CrossRefGoogle Scholar
[Pe1]Pesin, Ya.. Dimension type characteristics for invariant sets of dynamical systems. Russian Math. Surveys 43 (1988), 111–151.CrossRefGoogle Scholar
[Pe2]Pesin, Ya.. Dimension Theory in Dynamical Systems: Contemporary View and Applications. Chicago University Press, to appear.CrossRefGoogle Scholar
[PP]Pesin, Ya. and Pitskel, B.. Topological pressure and the variational principle for noncompact sets. Functional Anal. Appl. 18 (1984), 307–318.CrossRefGoogle Scholar
[PeW1]Pesin, Ya. and Weiss, H.. On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture. Comm. Math. Phys. To appear.Google Scholar
[PeW2]Pesin, Ya. and Weiss, H.. A multifractal analysis of Gibbs measures for conformal expanding maps and Markov Moran geometric constructions. Preprint. PSU, 1995.Google Scholar
[PoW]Pollicott, M. and Weiss, H.. The dimensions of some self affine limit sets in the plane and hyperbolic sets. J. Statist. Phys. 77 (1994), 841–866.CrossRefGoogle Scholar
[PU]Przytycki, F. and Urbański, M.. On the Hausdorff dimension of some fractal sets. Studia Mathematica 93 (1989), 155–186.CrossRefGoogle Scholar
[R1]Ruelle, D.. Statistical mechanics on a compact set with ․μ action satisfying expansiveness and specification. Trans. Amer. Math. Soc. 185 (1973), 237–251.CrossRefGoogle Scholar
[R2]Ruelle, D.. Thermodynamic Formalism (Encyclopedia of Mathematics and its Applications 5). Addison-Wesley, 1978.Google Scholar
[R3]Ruelle, D.. Repellers for real analytic maps. Ergod. Th. & Dynam. Sys. 2 (1982), 99–107.CrossRefGoogle Scholar
[SS]Schmeling, J. and Siegmund-Schultze, R.. Hölder continuity of the holonomy maps for hyperbolic sets I. Ergodic Theory and Related Topics III, Proc. Int. Conf. (Güstrow, Germany, October 22–27, 1990), (Lecture Notes in Mathematics 1514). Eds Krengel, U., Richter, K. and Warstat, V.. Springer, 1992, pp. 174–191.CrossRefGoogle Scholar
[T]Takens, F.. Limit capacity and Hausdorff dimension of dynamically defined Cantor sets. Dynamical Systems (Valparaiso 1986) (Lecture Notes in Mathematics 1331). Eds Bamón, R., Labarca, R. and Palis, J.Jr. Springer, 1988, pp. 196–212.CrossRefGoogle Scholar
[W]Walters, P.. A variational principle for the pressure of continuous transformations. Amer. J. Math. 97 (1976), 937–971.CrossRefGoogle Scholar