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
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