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 \(1 \pmod 4\) and their generalizations to orbit types of period \(n\) congruent to \(3 \pmod 4\) 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 \(0\pmod 4\) which attain maximum entropy amongst \(n\)-cycles.

Readership

Graduate students and research mathematicians interested in dynamical systems and ergodic theory.

Table of Contents

• Introduction
• Preliminaries
• Some useful properties of the induced matrix of a maximodal permutation
• The family of orbit types
• Some easy lemmas
• Two inductive lemmas
• The remaining case
• References