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 > is an element of X x 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 of N is called an independence set of {A(1), A(2), . . . , A(k)} if for any non-empty finite subset J of I and S is an element of {1, 2, . . . , k}(J), boolean AND(i is an element of J) f(-i)A(S(i)) not equal empty set. For a circle map or interval map (M, f), a pair < x, y > is an element of M x M with x not equal y is called non-separable if there exists z is an element of M such that x, y is an element of omega(z, f) and < x, y > can not be separated. For a circle map f : S -> S with zero topological entropy, we show that a non-diagonal pair < x, y > is an element of S x 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.