Zero entropy and stable rotation sets for monotone recurrence relations. (English) Zbl 1517.37027
Summary: 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 [Math. Z. 297, No. 3–4, 1673–1692 (2021; Zbl 1465.37024)]
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.
MSC:
| 37B40 | Topological entropy |
| 37B65 | Approximate trajectories, pseudotrajectories, shadowing and related notions for topological dynamical systems |
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
Citations:
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