Some properties of circle maps with zero topological entropy

Abstract

In this paper we introduce three pairs in ‘local entropy theory’. For a dynamical system (X, f), a pair is called an IN-pair (reps. an IT-pair) if for any neighborhoods U 1 and U 2 of x and y respectively, has arbitrarily large finite independence sets (reps. has an infinite independence set) where is called an independence set of if for any non-empty finite subset J of I and S ∈ {1, 2, …, k} J , ⋂ i∈J f −i A S(i) ≠ ∅. For a circle map or interval map (M, f), a pair ⟨x, y⟩ ∈ M × M with x ≠ y is called non-separable if there exists z ∈ M such that x, y ∈ ω(z, f) and ⟨x, y⟩ can not be separated. For a circle map with zero topological entropy, we show that a non-diagonal pair is non-separable if and only if it is an IN-pair if and only if it is an IT-pair. We introduce the maximal pattern entropy and recall that a null system is a system with zero maximal pattern entropy. We also show that if a circle map is topological null then the maximal pattern entropy of every open cover is of polynomial order.