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Measures of maximal and full dimension for smooth maps

Part of: Infinite-dimensional dissipative dynamical systems Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory

Published online by Cambridge University Press:  17 March 2023

YURONG CHEN ,
CHIYI LUO  and
YUN ZHAO
YURONG CHEN
Affiliation:
School of Mathematical Sciences, Soochow University, Suzhou 215006, Jiangsu, P. R. China and Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou 215006, Jiangsu, P. R. China (e-mail: 20184207019@stu.suda.edu.cn, luochiyi98@gmail.com)
CHIYI LUO
Affiliation:
School of Mathematical Sciences, Soochow University, Suzhou 215006, Jiangsu, P. R. China and Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou 215006, Jiangsu, P. R. China (e-mail: 20184207019@stu.suda.edu.cn, luochiyi98@gmail.com)
YUN ZHAO*
Affiliation:
School of Mathematical Sciences, Soochow University, Suzhou 215006, Jiangsu, P. R. China and Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou 215006, Jiangsu, P. R. China (e-mail: 20184207019@stu.suda.edu.cn, luochiyi98@gmail.com)
*
(e-mail: zhaoyun@suda.edu.cn)
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Abstract

For a $C^1$ non-conformal repeller, this paper proves that there exists an ergodic measure of full Carathéodory singular dimension. For an average conformal hyperbolic set of a $C^1$ diffeomorphism, this paper constructs a Borel probability measure (with support strictly inside the repeller) of full Hausdorff dimension. If the average conformal hyperbolic set is of a $C^{1+\alpha }$ diffeomorphism, this paper shows that there exists an ergodic measure of maximal dimension.

Keywords

measure of full dimensionmeasure of maximal dimensionrepellerhyperbolic sets

MSC classification

Primary: 37C45: Dimension theory of dynamical systems 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Secondary: 37L30: Attractors and their dimensions, Lyapunov exponents

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 1 , January 2024 , pp. 31 - 49
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Aoki, N.. Topological Dynamics . Topics in General Topology (North-Holland Mathematical Library, 41). Eds. K. Morita and J.-I. Nagata. North-Holland, Amsterdam, 1989, Ch. 15, pp. 625–740.10.1016/S0924-6509(08)70161-2CrossRefGoogle Scholar
Ban, J., Cao, Y. and Hu, H.. The dimensions of a non-conformal repeller and an average conformal repeller. Trans. Amer. Math. Soc. 362 (2010), 727–751.10.1090/S0002-9947-09-04922-8CrossRefGoogle Scholar
Barreira, L.. Nonadditive thermodynamic formalism: equilibrium and Gibbs measures. Discrete Contin. Dyn. Syst. 16 (2006), 279–305.10.3934/dcds.2006.16.279CrossRefGoogle Scholar
Barreira, L.. Dimension and Recurrence in Hyperbolic Dynamics (Progress in Mathematics, 272). Birkhäuser Verlag, Basel, 2008.Google Scholar
Barreira, L. and Wolf, C.. Measures of maximal dimension for hyperbolic diffeomorphisms. Comm. Math. Phys. 239(1) (2003), 93–113.10.1007/s00220-003-0858-9CrossRefGoogle Scholar
Barreira, L. and Wolf, C.. Pointwise dimension and ergodic decompositions. Ergod. Th. & Dynam. Sys. 26 (2006), 653–671.10.1017/S0143385705000672CrossRefGoogle Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Springer-Verlag, Berlin, 1975.10.1007/BFb0081279CrossRefGoogle Scholar
Bowen, R.. Hausdorff dimension of quasicircles. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 11–25.10.1007/BF02684767CrossRefGoogle Scholar
Cao, Y., Feng, D. and Huang, W.. The thermodynamic formalism for sub-additive potentials. Discrete Contin. Dyn. Syst. 20 (2008), 639–657.10.3934/dcds.2008.20.639CrossRefGoogle Scholar
Cao, Y., Hu, H. and Zhao, Y.. Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure. Ergod. Th. & Dynam. Sys. 33 (2013), 831–850.10.1017/S0143385712000090CrossRefGoogle Scholar
Cao, Y., Pesin, Y. and Zhao, Y.. Dimension estimates for non-conformal repellers and continuity of sub-additive topological pressure. Geom. Funct. Anal. 29 (2019), 1325–1368.10.1007/s00039-019-00510-7CrossRefGoogle Scholar
Cao, Y., Wang, J. and Zhao, Y.. Dimension approximation in smooth dynamical systems. Preprint, 2023, arXiv:2301.06233.10.1017/etds.2023.26CrossRefGoogle Scholar
Chung, Y.. The largeness of sets of points with non-dense orbit in basic sets on surfaces. Proc. Amer. Math. Soc. 124(5) (1996), 1615–1624.10.1090/S0002-9939-96-03215-7CrossRefGoogle Scholar
Denker, M. and Urbański, M.. On Sullivan’s conformal measures for rational maps of the Riemann sphere. Nonlinearity 4(2) (1991), 365–384.10.1088/0951-7715/4/2/008CrossRefGoogle Scholar
Falconer, K.. Dimensions and measures of quasi self-similar sets. Proc. Amer. Math. Soc. 106 (1989), 543–554.10.1090/S0002-9939-1989-0969315-8CrossRefGoogle Scholar
Fang, J., Cao, Y. and Zhao, Y.. Measure theoretic pressure and dimension formula for non-ergodic measures. Discrete Contin. Dyn. Syst. 40(5) (2020), 2767–2789.10.3934/dcds.2020149CrossRefGoogle Scholar
Feng, D. and Huang, W.. Variational principle for weighted topological pressure. J. Math. Pures Appl. (9) 106 (2016), 411–452.10.1016/j.matpur.2016.02.016CrossRefGoogle Scholar
Gatzouras, D. and Peres, Y.. Invariant measures of full dimension for some expanding maps. Ergod. Th. & Dynam. Sys. 17 (1997), 147–167.10.1017/S0143385797060987CrossRefGoogle Scholar
Morris, I.. Mather sets for sequences of matrices and applications to the study of joint spectral radii. Proc. Lond. Math. Soc. (3) 107(1) (2013), 121–150.10.1112/plms/pds080CrossRefGoogle Scholar
Oseledec, V. I.. A multiplicative ergodic theorem. Characteristic Lyapnov exponents of dynamical systems. Tr. Moskov. Mat. Obšž 19 (1968), 179–210.Google Scholar
Pesin, Y.. Dimension Theory in Dynamical Systems: Contemporary Views and Applications. University of Chicago Press, Chicago, IL, 1997.10.7208/chicago/9780226662237.001.0001CrossRefGoogle Scholar
Rams, M.. Measures of maximal dimension for linear horseshoes. Real Anal. Exchange 31(1) (2005/2006), 55–62.10.14321/realanalexch.31.1.0055CrossRefGoogle Scholar
Ruelle, D.. Thermodynamic Formalism. Addison-Wesley Publishing Co., Boston, MA, 1978.Google Scholar
Ruelle, D.. Repellers for real analytic maps. Ergod. Th. & Dynam. Sys. 2 (1982), 99–107.10.1017/S0143385700009603CrossRefGoogle Scholar
Urbański, M.. Measures and dimensions in conformal dynamics. Bull. Amer. Math. Soc. (N.S.) 40 (2003), 281–321.10.1090/S0273-0979-03-00985-6CrossRefGoogle Scholar
Urbański, M. and Wolf, C.. Ergodic theory of parabolic horseshoes, Comm. Math. Phys. 281(3) (2008), 711–751.10.1007/s00220-008-0498-1CrossRefGoogle Scholar
Walters, P.. An Introduction to Ergodic Theory. Springer-Verlag, New York, 1982.10.1007/978-1-4612-5775-2CrossRefGoogle Scholar
Wang, J. and Cao, Y.. The Hausdorff dimension estimation for an ergodic hyperbolic measure of ${C}^1$ -diffeomorphism. Proc. Amer. Mathw. Soc. 144(1) (2016), 119–128.10.1090/proc/12696CrossRefGoogle Scholar
Wang, J., Wang, J., Cao, Y. and Zhao, Y.. Dimensions of ${C}^1$ -average conformal hyperbolic sets Discrete Contin. Dyn. Syst. 40(2) (2020), 883–905.10.3934/dcds.2020065CrossRefGoogle Scholar
Wolf, C.. On measures of maximal and full dimension for polynomial automorphisms of ${C}^2$ . Trans. Amer. Math. Soc. 355(08) (2003), 3227–3239.10.1090/S0002-9947-03-03287-2CrossRefGoogle Scholar
Young, L. S.. Dimension, entropy and Lyapunov exponents. Ergod. Th. & Dynam. Sys. 2 (1982), 109–129.10.1017/S0143385700009615CrossRefGoogle Scholar