Maximum entropy of cycles of even period (Q2734766)

This monograph addresses the question of finding \(n\)-permutations and \(n\)-cycles which attain maximum entropy among all \(n\)-permutations and \(n\)-cycles. The question is motivated by the observation that a finite fully invariant set of a continuous map of the interval induces a permutation of that invariant set. If that permutation is a cycle, it is called the orbit type of the invariant set. \textit{M. Misiurewicz} and \textit{Z. Nitecki} [Mem. Am. Math. Soc. 456 (1991; Zbl 0745.58019)] showed that the maximum entropy for \(n\)-permutations and \(n\)-cycles is asymptotic to \(\log(2n/ \pi)\). To do this, they defined a family of \(n\)-cycles for \(n\) congruent to \(1\pmod 4\). \textit{W. Geller} and \textit{J. J. Tolosa} [Trans. Am. Math. Soc. 329, 161-171 (1992; Zbl 0751.58017)] extended this definition to all odd \(n\), and proved that this family actually attains maximum entropy among all \(n\)-permutations. The authors of this work construct a family of \(n\)-cycles which attain maximum entropy among all \(n\)-cycles whose period is congruent to \(0\pmod 4\).