# American Institute of Mathematical Sciences

June  2008, 21(2): 367-392. doi: 10.3934/dcds.2008.21.367

## Entropy formula for endomorphisms: Relations between entropy, exponents and dimension

 1 School of Mathematical Sciences, Peking University, Beijing 100871, China 2 School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  February 2007 Revised  December 2007 Published  March 2008

We present an entropy formula of Ledrappier-Young type for invariant measures (maybe non-SRB) of $C^2$ endomorphisms (maybe non-invertible and with singularities) on a compact manifold via their inverse limit spaces. This result may be considered as the most general form of entropy formula for a deterministic system with an invariant measure, and a preliminary step to Eckmann-Ruelle conjecture. As an important application, we have proved the exact dimensionality of ergodic measures invariant under expanding maps.
Citation: Min Qian, Jian-Sheng Xie. Entropy formula for endomorphisms: Relations between entropy, exponents and dimension. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 367-392. doi: 10.3934/dcds.2008.21.367
 [1] Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281 [2] Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935 [3] Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 [4] Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

2019 Impact Factor: 1.338