Topological entropy and chaos of interval maps

Abstract

IN THIS paper we generalize the method introduced in [6] and use it to prove two results. As for another application of this generalized method which is on the number of periodic points, see [7]. The first result can be described as follows. Let I be a compact real interval and let f E CO(I, r> have a periodic point of minimal period 2n + 1 with integer n > 0. Then there exists a closed subinterval K, of I and, for every positive integer k, there are at least dk,, pairwise disjoint closed subintervals Jikn’s of K,, such that fk(Jlk,) > K,,. Furthermore, these numbers dk., are explicitly determined and have the property that ieX(log d,.,)/k = log &. where A,, is