Maximizing entropy of cycles on trees (Q379493)

Let \(f:T\to T\) be a continuous map of a tree \(T\), and \(P\subseteq T\) be an \(n\)-periodic orbit of \(f\). Then the triple \((T,P,f)\) is called an \(n\)-periodic model. A pattern \(\mathcal P\) is an equivalence class of \(n\)-periodic models, where the equivalence is defined via the so-called discrete components. There, a discrete component of \((T,P)\) is a subset \(Q\) of \(P\) such that either \(Q=P\) is a singleton, or \(|Q|>1\) and \(Q\) is the intersection of \(P\) with the closure of a connected component of \(T\setminus P\).NEWLINENEWLINEThe topological entropy \(h(\mathcal P)\) of a pattern \(\mathcal P\) is the infimum of topological entropies \(h(f)\) of corresponding models \((T,P,f)\). To study the entropy of a pattern it suffices to deal with monotone models; thus \(h(\mathcal P)\) can be easily computed as the logarithm of the spectral radius of the path transition matrix.NEWLINENEWLINEFor given \(n\), an \(n\)-periodic pattern is said to be maximal if its entropy is maximal among all \(n\)-periodic patterns. The main result of the paper states that every maximal pattern \(\mathcal P\) is simplicial, irreducible, and maximodal. That is, for a monotone model \((T,P,f)\) of \(\mathcal P\), every discrete component has two points, the path transition matrix is irreducible, and there is no triple \((a,x,b)\) of distinct points from \(P\) such that \([a,b]\cap P=\{a,x,b\}\) and \(f|_{[a,b]}\) is monotone.NEWLINENEWLINEThe proof of this theorem starts with showing that, among maximal \(n\)-periodic patterns, there always exists a simplicial one (Corollary 4.2). By Corollaries~5.4 and 6.2, maximal simplicial patterns are irreducible and maximodal. The most difficult part of the proof is to show Theorem~7.1 stating that every maximal pattern is simplicial. As a result of independent interest, the authors prove in Theorem 5.3 that, for every \(n\geq 4\), there is an \(n\)-periodic interval pattern with entropy larger than or equal to \(\log\lfloor n/2\rfloor\).