Skip to main content Accessibility help

Bounded distortion and dimension for non-conformal repellers

  • K. J. Falconer (a1) (a2)


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.



Hide All
[1]Bedford, T.. The box dimension of self-affine graphs and repellers. Nonlinearity 2 (1989), 5371.
[2]Bedford, T.. Applications of dynamical systems to fractals, in Fractal Geometry and Analysis (eds. Bélair, J. and Debuc, S.) 144 (Kluwer, 1991).
[3]Bedford, T. and Urbanski, M.. The box and Hausdorff dimension of self-affine sets. Ergod. Theory Dyn. Syst. 10 (1990), 627644.
[4]Bowen, R.. Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics 470 (Springer Verlag, 1978).
[5]Bowen, R.. Hausdorff dimension of quasi-circles. Publ. Math. IHES 50 (1979), 259273.
[6]Constantin, P., Foias, C. and Temam, R.. Attractors representing turbulent flows. Mem. Amer. Math. Soc. 53 (1985), No. 314.
[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), 327336.
[8]Douady, A. and Oesterlé, J.. Dimension de Hausdorff des attractors. C.R. Acad. Sci. Paris Sér. I Math. 290 (1980), 11351138.
[9]Falconer, K. J.. The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Phil. Soc. 103 (1988), 339350.
[10]Falconer, K. J.. A subadditive thermodynamic formalism for mixing repellers. J. Phys. A 21 (1988), L737–L742.
[11]Falconer, K. J.. Fractal Geometry, Mathematical Foundations and Applications (John Wiley, 1990).
[12]Falconer, K. J.. The dimension of self-affine fractals II. Math. Proc. Cambridge Phil. Soc. 111 (1992), 169179.
[13]Fathi, A.. Expansiveness, hyperbolicity and Hausdorff dimension. Commun. Math. Phys. 126 (1989), 249262.
[14]Gu, X.. An upper bound for the Hausdorff dimension of a hyperbolic set. Nonlinearity 4 (1991), 927934.
[15]Lloyd, N.. Degree Theory (Cambridge University Press, 1978).
[16]McMullen, C.. The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96 (1984) 19.
[17]Ruelle, D.. Thermodynamic Formalism (Addison Wesley, 1978).
[18]Ruelle, D.. Repellers for real analytic maps. Ergod. Theory Dyn. Syst. 3 (1982), 99108.
[19]Sinai, Ya. G.. Construction of Markov partitions. Fund. Anal. Appl. 2 (1968), 245253.
[20]Walters, P.. An Introduction to Ergodic Theory (Springer, 1982).
[21]Ledrappier, F. and Young, L.-S., Dimension formula for random transformations. Commun. Math. Phys. 117 (1988), 217240.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed