Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T14:15:36.808Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounded distortion and dimension for non-conformal repellers

Published online by Cambridge University Press:  24 October 2008

K. J. Falconer
K. J. Falconer
Affiliation:
School of Mathematics, University Walk, Bristol BS8 1TW Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS*
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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
[15]Lloyd, N.. Degree Theory (Cambridge University Press, 1978).Google Scholar
[16]McMullen, C.. The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96 (1984) 1–9.CrossRefGoogle Scholar
[17]Ruelle, D.. Thermodynamic Formalism (Addison Wesley, 1978).Google 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
[20]Walters, P.. An Introduction to Ergodic Theory (Springer, 1982).CrossRefGoogle Scholar
[21]Ledrappier, F. and Young, L.-S., Dimension formula for random transformations. Commun. Math. Phys. 117 (1988), 217–240.CrossRefGoogle Scholar