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.

在本文中，我们介绍了“局部熵理论”中的三对。对于动力系统（X，˚F），一对被称为IN -pair（代表。一个IT -pair）如果由于任何邻域ù ₁和ü ₂的X和ÿ分别具有任意大的有限独立集（代表。具有无限的独立集合），其中被称为独立组，如果对于任何非空有限子集Ĵ的我和小号∈{1，2，...，ķ } ^Ĵ，⋂_{我 ∈ J} f ^{− i} A _{S ( i )} ≠ ∅。对于圆图或区间图 ( M , f )，如果存在z ∈ M使得x , y ∈ ω ( z , f )有x ≠ y 的一对 ⟨ x , y ⟩ ∈ M M称为不可分和 ⟨ x , y ⟩ 不能分开。对于圆形地图 在拓扑熵为零的情况下，我们证明了一个非对角对是不可分的当且仅当它是一个IN对当且仅当它是一个IT对。我们介绍了最大模式熵，并记得零系统是一个最大模式熵为零的系统。我们还表明，如果圆图是拓扑空的，那么每个开盖的最大模式熵是多项式的。