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Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes

Published online by Cambridge University Press:  07 November 2017

CHENG CHENG ,
SYLVAIN CROVISIER ,
SHAOBO GAN ,
XIAODONG WANG  and
DAWEI YANG
CHENG CHENG
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, PR China email chocolate-74@163.com, gansb@pku.edu.cn
SYLVAIN CROVISIER
Affiliation:
CNRS - Laboratoire de Mathématiques d’Orsay, Université Paris-Sud 11, Orsay 91405, France email Sylvain.Crovisier@math.u-psud.fr
SHAOBO GAN
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, PR China email chocolate-74@163.com, gansb@pku.edu.cn
XIAODONG WANG
Affiliation:
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, PR China email xdwang1987@sjtu.edu.cn, xdwang1987@gmail.com
DAWEI YANG
Affiliation:
School of Mathematical Sciences, Soochow University, Suzhou 215006, PR China email yangdw1981@gmail.com, yangdw@suda.edu.cn
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Abstract

We prove that, for $C^{1}$-generic diffeomorphisms, if a homoclinic class is not hyperbolic, then there is a non-trivial non-hyperbolic ergodic measure supported on it. This proves a conjecture by Díaz and Gorodetski.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 7 , July 2019 , pp. 1805 - 1823
Copyright
© Cambridge University Press, 2017 

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