Interval maps of given topological entropy and Sharkovskii's type

Abstract

It is known that the topological entropy of a continuous interval map f is positive if and only if the type of f for Sharkovskii's order is 2^d p for some odd integer p ≥ 3 and some d ≥ 0; and in this case the topological entropy of f is greater than or equal to log(λ_p) / 2^d, where λ_p is the unique positive root of X^p - 2X^p-2 - 1. For every odd p ≥ 3, every d ≥ 0 and every λ ≥ λ_p, we build a piecewise monotone continuous interval map that is of type 2^d p for Sharkovskii's order and whose topological entropy is log (λ) / 2^d. This shows that, for a given type, every possible finite entropy above the minimum can be reached provided the type allows the map to have positive entropy. Moreover, if d = 0 the map we build is topologically mixing.

Paper: [arXiv:1906.03649] [pdf]