Some properties of circle maps with zero topological entropy

In this paper we introduce three pairs in 'local entropy theory'. For a dynamical system (X, f), a pair $⟨x,y⟩\in X\times X$ is called an IN-pair (reps. an IT-pair) if for any neighborhoods U ₁ and U ₂ of x and y respectively, $\{U_1,U_2\}$ has arbitrarily large finite independence sets (reps. $\{U_1,U_2\}$ has an infinite independence set) where $I\subset N$ is called an independence set of $\{A_1,A_2,\dots ,A_k\}$ 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 $f:S\to S$ with zero topological entropy, we show that a non-diagonal pair $⟨x,y⟩\in S\times S$ 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. (paper)