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Dimension Theory for Invariant Measures of Endomorphisms

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Abstract

We establish the exact dimensional property of an ergodic hyperbolic measure for a C 2 non-invertible but non-degenerate endomorphism on a compact Riemannian manifold without boundary. Based on this, we give a new formula of Lyapunov dimension of ergodic measures and show it coincides with the dimension of hyperbolic ergodic measures in a setting of random endomorphisms. Our results extend several well known theorems of Barreira et al. (Ann Math 149:755–783, 1999) and Ledrappier and Young [Commun Math Phys 117(4):529–548, 1988] for diffeomorphisms to the case of endomorphisms.

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Authors and Affiliations

  1. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China

    Lin Shu

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  1. Lin Shu
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Correspondence to Lin Shu.

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Communicated by G. Gallavotti

This work is supported by NSFC (No. 10901007) and National Basic Research Program of China (973 Program) (2007 CB 814800).

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Shu, L. Dimension Theory for Invariant Measures of Endomorphisms. Commun. Math. Phys. 298, 65–99 (2010). https://doi.org/10.1007/s00220-010-1059-y

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