Maximum entropy of cycles of even period

Abstract

A finite fully invariant set of a continuous map of the interval induces a permutation of that invariant set. If the permutation is a cycle, it is called its orbit type. It is known that Misiurewicz-Nitecki orbit types of period n congruent to 1 (mod 4) and their generalizations to orbit types of period n congruent to 3 (mod 4) have maximum entropy amongst all orbit types of odd period n and indeed amongst all n-permutations for n odd. We construct a family of orbit types of period n congruent to 0 (mod 4) which attain maximum entropy amongst n-cycles.