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A mechanism for ejecting a horseshoe from a partially hyperbolic chain recurrence class

Part of: Local and nonlocal bifurcation theory Dynamical systems with hyperbolic behavior Topological dynamics

Published online by Cambridge University Press:  03 November 2023

CHRISTIAN BONATTI  and
KATSUTOSHI SHINOHARA
CHRISTIAN BONATTI
Affiliation:
Institut de Mathématiques de Bourgogne CNRS – URM 5584, Université de Bourgogne, Dijon 21004, France (e-mail: christian.bonatti@u-bourgogne.fr)
KATSUTOSHI SHINOHARA*
Affiliation:
Graduate School of Business Administration, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan
*
e-mail: ka.shinohara@r.hit-u.ac.jp
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Abstract

We give a $C^1$-perturbation technique for ejecting an a priori given finite set of periodic points preserving a given finite set of homo/heteroclinic intersections from a chain recurrence class of a periodic point. The technique is first stated under a simpler setting called a Markov iterated function system, a two-dimensional iterated function system in which the compositions are chosen in a Markovian way. Then we apply the result to the setting of three-dimensional partially hyperbolic diffeomorphisms.

Keywords

iterated function systemswild diffeomorphismchain recurrence classC1-generic diffeomorphisms

MSC classification

Primary: 37B25: Lyapunov functions and stability; attractors, repellers
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37G35: Attractors and their bifurcations

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 8 , August 2024 , pp. 2080 - 2142
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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