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

Min Qian; Jian-Sheng Xie

doi:10.3934/dcds.2008.21.367

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2008, Volume 21, Issue 2: 367-392. Doi: 10.3934/dcds.2008.21.367

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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
2.
School of Mathematical Sciences, Fudan University, Shanghai 200433

Received: February 2007

Revised: December 2007

Published: May 2008

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Abstract

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.

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Mathematics Subject Classification: Primary: 37C40, 37C45; Secondary: 37D20.

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