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The Pesin entropy formula for $C^1$ diffeomorphisms with dominated splitting

Published online by Cambridge University Press:  26 February 2014

ELEONORA CATSIGERAS ,
MARCELO CERMINARA  and
HEBER ENRICH
ELEONORA CATSIGERAS
Affiliation:
Instituto de Matemática y Estadística Rafael Laguardia (IMERL), Facultad de Ingeniería, Universidad de la República, Uruguay email eleonora@fing.edu.uy, cerminar@fing.edu.uy, enrich@fing.edu.uy
MARCELO CERMINARA
Affiliation:
Instituto de Matemática y Estadística Rafael Laguardia (IMERL), Facultad de Ingeniería, Universidad de la República, Uruguay email eleonora@fing.edu.uy, cerminar@fing.edu.uy, enrich@fing.edu.uy
HEBER ENRICH
Affiliation:
Instituto de Matemática y Estadística Rafael Laguardia (IMERL), Facultad de Ingeniería, Universidad de la República, Uruguay email eleonora@fing.edu.uy, cerminar@fing.edu.uy, enrich@fing.edu.uy
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Abstract

For any $C^1$ diffeomorphism with dominated splitting, we consider a non-empty set of invariant measures that describes the asymptotic statistics of Lebesgue-almost all orbits. They are the limits of convergent subsequences of averages of the Dirac delta measures supported on those orbits. We prove that the metric entropy of each of these measures is bounded from below by the sum of the Lyapunov exponents on the dominating sub-bundle. As a consequence, if those exponents are non-negative, and if the exponents on the dominated sub-bundle are non-positive, those measures satisfy the Pesin entropy formula.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 3 , May 2015 , pp. 737 - 761
Copyright
© Cambridge University Press, 2014 

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