Dimensions of $ C^1- $average conformal hyperbolic sets

Juan Wang, ,School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, China, Jing Wang, Yongluo Cao, Yun Zhao, ,Departament of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200062, China, ,Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, China, ,Center for Dynamical Systems and Differential Equation, Soochow University, Suzhou 215006, Jiangsu, China
2020 Discrete and Continuous Dynamical Systems. Series A  
This paper introduces the concept of average conformal hyperbolic sets, which admit only one positive and one negative Lyapunov exponents for any ergodic measure. For an average conformal hyperbolic set of a C 1 diffeomorphism, utilizing the techniques in sub-additive thermodynamic formalism and some geometric arguments with unstable/stable manifolds, a formula of the Hausdorff dimension and lower (upper) box dimension is given in this paper, which is exactly the sum of the dimensions of the
more » ... imensions of the restriction of the hyperbolic set to stable and unstable manifolds. Furthermore, the dimensions of an average conformal hyperbolic set vary continuously with respect to the dynamics. 2010 Mathematics Subject Classification. Primary: 37C45, 37D20; Secondary: 37D35.
doi:10.3934/dcds.2020065 fatcat:folhh7ivdbglni3nnslatpb5ei