American Institute of Mathematical Sciences

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

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

Min Qian 1, and  Jian-Sheng Xie 2,

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.
Keywords: Lyapunov Exponents, Entropy, SRB Measure..
Mathematics Subject Classification: Primary: 37C40, 37C45; Secondary: 37D2.
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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