Positive Topological Entropy For Monotone Recurrence Relations

We associate the topological entropy of monotone recurrence relations with theAubry-Mather theory. If there exists an interval [ρ0, ρ1], such that for each ω∈(ρ0, ρ1),all Birkhof minimizers with rotation number ω do not form a foliation, then the d-ifeomorphism on the high-dimensional cylinder defined via the monotone recurrencerelation has positive topological entropy. Therefore, we need to proof the followingresults: Firstly, for monotone recurrence relation, each minimizer with bounded actionis Birkhof. Secondly, we construct in configuration space with bounded action two so-lutions exchanging rotation numbers and hence arrive at the conclusion by Angenent’scriterion in [1].