Positive topological entropy for monotone recurrence relations

Abstract

We associate the topological entropy of monotone recurrence relations with the Aubry-Mather theory. If there exists an interval [ρ 0 , ρ 1 ] such that, for each ω ∈ (ρ 0 , ρ 1 ), all Birkhoff minimizers with rotation number ω do not form a foliation, then the diffeomorphism on the high-dimensional cylinder defined via the monotone recurrence relation has positive topological entropy.

Minimizers with Bounded Action for the High-Dimensional Frenkel–Kontorova Model

“…It was shown by Mramor and Rink [2013] that under some conditions, each minimizer is either Birkhoff, or it is oscillating and exponentially growing. We proved in [Guo et al, 2014] that this conclusion holds true even if we relax the strong twist condition in [Mramor & Rink, 2013]. Precisely, we showed that for the one-dimensional generalized FK model each minimizer with bounded action is Birkhoff.…”

“…We should mention that for high-dimensional FK model, whether the minimizer x itself is Birkhoff is still unclear, although it is for the onedimensional case, see [Guo et al, 2014;Mramor & Rink, 2013].…”

Periodic Generalized Birkhoff Solutions and Farey Intervals for Monotone Recurrence Relations

“…It should be pointed out that Proposition 2.5 has provided great convenience to determine whether the system has positive topological entropy, which has been applied in [2,3,16,21,22,24] under different background and conditions.…”

Zero entropy and stable rotation sets for monotone recurrence relations

“…If the monotone recurrence relation of equation (1.1) has a generating function, then zero topological entropy implies that Birkhoff minimizers with each rotation number form a continuous foliation [19]. Our first topic in this paper is to investigate, for the general monotone recurrence relations, the properties of rotation sets of equation (1.1) with zero topological entropy.…”