The dimensions of a non-conformal repeller and an average conformal repeller

Ban, Jungchao; Cao, Yongluo; Hu, Huyi

doi:10.1090/S0002-9947-09-04922-8

The dimensions of a non-conformal repeller and an average conformal repeller
HTML articles powered by AMS MathViewer

by Jungchao Ban, Yongluo Cao and Huyi Hu PDF

Trans. Amer. Math. Soc. 362 (2010), 727-751 Request permission

Abstract:

In this paper, using thermodynamic formalism for the sub-additive potential, upper bounds for the Hausdorff dimension and the box dimension of non-conformal repellers are obtained as the sub-additive Bowen equation. The map $f$ only needs to be $C^1$, without additional conditions. We also prove that all the upper bounds for the Hausdorff dimension obtained in earlier papers coincide. This unifies their results. Furthermore we define an average conformal repeller and prove that the dimension of an average conformal repeller equals the unique root of the sub-additive Bowen equation.

References

Luis Barreira, Dimension estimates in nonconformal hyperbolic dynamics, Nonlinearity 16 (2003), no. 5, 1657–1672. MR 1999573, DOI 10.1088/0951-7715/16/5/307
Luis M. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergodic Theory Dynam. Systems 16 (1996), no. 5, 871–927. MR 1417767, DOI 10.1017/S0143385700010117
Rufus Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11–25. MR 556580
Yong-Luo Cao, De-Jun Feng, and Wen Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst. 20 (2008), no. 3, 639–657. MR 2373208, DOI 10.3934/dcds.2008.20.639
Adrien Douady and Joseph Oesterlé, Dimension de Hausdorff des attracteurs, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 24, A1135–A1138 (French, with English summary). MR 585918
K. J. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 2, 339–350. MR 923687, DOI 10.1017/S0305004100064926
K. J. Falconer, Dimensions and measures of quasi self-similar sets, Proc. Amer. Math. Soc. 106 (1989), no. 2, 543–554. MR 969315, DOI 10.1090/S0002-9939-1989-0969315-8
K. J. Falconer, Bounded distortion and dimension for nonconformal repellers, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 2, 315–334. MR 1277063, DOI 10.1017/S030500410007211X
Dimitrios Gatzouras and Yuval Peres, Invariant measures of full dimension for some expanding maps, Ergodic Theory Dynam. Systems 17 (1997), no. 1, 147–167. MR 1440772, DOI 10.1017/S0143385797060987
Huyi Hu, Dimensions of invariant sets of expanding maps, Comm. Math. Phys. 176 (1996), no. 2, 307–320. MR 1374415
F. Ledrappier, Some relations between dimension and Lyapounov exponents, Comm. Math. Phys. 81 (1981), no. 2, 229–238. MR 632758
Yakov B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. Contemporary views and applications. MR 1489237, DOI 10.7208/chicago/9780226662237.001.0001
David Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 99–107. MR 684247, DOI 10.1017/s0143385700009603
David Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat. 9 (1978), no. 1, 83–87. MR 516310, DOI 10.1007/BF02584795
Károly Simon, Hausdorff dimension of hyperbolic attractors in ${\Bbb R}^3$, Fractal geometry and stochastics III, Progr. Probab., vol. 57, Birkhäuser, Basel, 2004, pp. 79–92. MR 2087133
Károly Simon and Boris Solomyak, Hausdorff dimension for horseshoes in $\mathbf R^3$, Ergodic Theory Dynam. Systems 19 (1999), no. 5, 1343–1363. MR 1721625, DOI 10.1017/S0143385799141671
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
Yingjie Zhang, Dynamical upper bounds for Hausdorff dimension of invariant sets, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 739–756. MR 1452191, DOI 10.1017/S0143385797085003