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Shub’s example revisited

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  23 September 2024

CHAO LIANG ,
RADU SAGHIN ,
FAN YANG  and
JIAGANG YANG
CHAO LIANG
Affiliation:
School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 100081, China (e-mail: chaol@cufe.edu.cn)
RADU SAGHIN*
Affiliation:
Instituto de Matemática, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile
FAN YANG
Affiliation:
Department of Mathematics, Wake Forest University, Winston-Salem, NC, USA (e-mail: yangf@wfu.edu)
JIAGANG YANG
Affiliation:
Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Brazil (e-mail: yangjg@impa.br)
*
e-mail: radu.saghin@pucv.cl
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Abstract

For a class of robustly transitive diffeomorphisms on ${\mathbb T}^4$ introduced by Shub [Topologically transitive diffeomorphisms of $T^4$. Proceedings of the Symposium on Differential Equations and Dynamical Systems (Lecture notes in Mathematics, 206). Ed. D. Chillingworth. Springer, Berlin, 1971, pp. 39–40], satisfying an additional bunching condition, we show that there exists a $C^2$ open and $C^r$ dense subset ${\mathcal U}^r$, $2\leq r\leq \infty $, such that any two hyperbolic points of $g\in {\mathcal U}^r$ with stable index $2$ are homoclinically related. As a consequence, every $g\in {\mathcal U}^r$ admits a unique homoclinic class associated to the hyperbolic periodic points with index $2$, and this homoclinic class coincides with the whole ambient manifold. Moreover, every $g\in {\mathcal U}^r$ admits at most one measure of maximal entropy, and every $g\in {\mathcal U}^{\infty }$ admits a unique measure of maximal entropy.

Keywords

partially hyperbolic diffeomorphismshomoclinic classesequilibrium states

MSC classification

Primary: 37D30: Partially hyperbolic systems and dominated splittings 37C40: Smooth ergodic theory, invariant measures
Secondary: 37D35: Thermodynamic formalism, variational principles, equilibrium states 37C29: Homoclinic and heteroclinic orbits

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 3 , March 2025 , pp. 894 - 914
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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