Article contents
Bounded distortion and dimension for non-conformal repellers
Published online by Cambridge University Press: 24 October 2008
Abstract
We obtain an expression for the dimension of a mixing repeller of a non-conformal mapping analogous to the well-known Bowen-Ruelle formula for conformal repellers. The dimension is given in terms of a generalized pressure defined in the context of the thermodynamic formalism. In the course of the paper we develop a subadditive version of the thermodynamic formalism that is suited to our needs and also obtain a ‘bounded distortion’ principle applicable to the non-conformal situation.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 2 , March 1994 , pp. 315 - 334
- Copyright
- Copyright © Cambridge Philosophical Society 1994
References
REFERENCES
[1]Bedford, T.. The box dimension of self-affine graphs and repellers. Nonlinearity 2 (1989), 53–71.CrossRefGoogle Scholar
[2]Bedford, T.. Applications of dynamical systems to fractals, in Fractal Geometry and Analysis (eds. Bélair, J. and Debuc, S.) 1–44 (Kluwer, 1991).Google Scholar
[3]Bedford, T. and Urbanski, M.. The box and Hausdorff dimension of self-affine sets. Ergod. Theory Dyn. Syst. 10 (1990), 627–644.CrossRefGoogle Scholar
[4]Bowen, R.. Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics 470 (Springer Verlag, 1978).Google Scholar
[5]Bowen, R.. Hausdorff dimension of quasi-circles. Publ. Math. IHES 50 (1979), 259–273.CrossRefGoogle Scholar
[6]Constantin, P., Foias, C. and Temam, R.. Attractors representing turbulent flows. Mem. Amer. Math. Soc. 53 (1985), No. 314.Google Scholar
[7]Deliu, A., Geronimo, J. S., Shonkwiler, R. and Hardin, D.. Dimensions associated with recurrent self-similar sets. Math. Proc. Cambridge Phil. Soc. 110 (1991), 327–336.CrossRefGoogle Scholar
[8]Douady, A. and Oesterlé, J.. Dimension de Hausdorff des attractors. C.R. Acad. Sci. Paris Sér. I Math. 290 (1980), 1135–1138.Google Scholar
[9]Falconer, K. J.. The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Phil. Soc. 103 (1988), 339–350.CrossRefGoogle Scholar
[10]Falconer, K. J.. A subadditive thermodynamic formalism for mixing repellers. J. Phys. A 21 (1988), L737–L742.CrossRefGoogle Scholar
[11]Falconer, K. J.. Fractal Geometry, Mathematical Foundations and Applications (John Wiley, 1990).Google Scholar
[12]Falconer, K. J.. The dimension of self-affine fractals II. Math. Proc. Cambridge Phil. Soc. 111 (1992), 169–179.CrossRefGoogle Scholar
[13]Fathi, A.. Expansiveness, hyperbolicity and Hausdorff dimension. Commun. Math. Phys. 126 (1989), 249–262.CrossRefGoogle Scholar
[14]Gu, X.. An upper bound for the Hausdorff dimension of a hyperbolic set. Nonlinearity 4 (1991), 927–934.CrossRefGoogle Scholar
[16]McMullen, C.. The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96 (1984) 1–9.CrossRefGoogle Scholar
[18]Ruelle, D.. Repellers for real analytic maps. Ergod. Theory Dyn. Syst. 3 (1982), 99–108.CrossRefGoogle Scholar
[19]Sinai, Ya. G.. Construction of Markov partitions. Fund. Anal. Appl. 2 (1968), 245–253.CrossRefGoogle Scholar
[21]Ledrappier, F. and Young, L.-S., Dimension formula for random transformations. Commun. Math. Phys. 117 (1988), 217–240.CrossRefGoogle Scholar
- 54
- Cited by