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Zero entropy and stable rotation sets for monotone recurrence relations

Part of: Low-dimensional dynamical systems Smooth dynamical systems: general theory Infinite-dimensional dissipative dynamical systems Topological dynamics

Published online by Cambridge University Press:  05 April 2022

WEN-XIN QIN ,
BAI-NIAN SHEN ,
YI-LIN SUN  and
TONG ZHOU
WEN-XIN QIN*
Affiliation:
Department of Mathematics, Soochow University, Suzhou 215006, China
BAI-NIAN SHEN
Affiliation:
Department of Mathematics, Soochow University, Suzhou 215006, China
YI-LIN SUN
Affiliation:
Department of Mathematics, Soochow University, Suzhou 215006, China
TONG ZHOU
Affiliation:
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, China
*
e-mail: qinwx@suda.edu.cn
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Abstract

In this paper, we show that each element in the convex hull of the rotation set of a compact invariant chain transitive set is realized by a Birkhoff solution, which is an improvement of the fundamental lemma of T. Zhou and W.-X. Qin [Pseudo solutions, rotation sets, and shadowing rotations for monotone recurrence relations. Math. Z. 297 (2021), 1673–1692] in the study of rotation sets for monotone recurrence relations. We then investigate the properties of rotation sets assuming the system has zero topological entropy. The rotation set for a Birkhoff recurrence class is a singleton and the forward and backward rotation numbers are identical for each solution in the same Birkhoff recurrence class. We also show the continuity of rotation numbers on the set of non-wandering points. If the rotation set is upper-stable, then we show that each boundary point is a rational number, and we also obtain a result of bounded deviation.

Keywords

rotation settopological entropychain transitivitymonotone recurrence relation

MSC classification

Primary: 37B40: Topological entropy 37C65: Monotone flows 37E40: Twist maps 37E45: Rotation numbers and vectors 37L60: Lattice dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 5 , May 2023 , pp. 1737 - 1759
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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