Fair dice-like hyperbolic systems (Q2906451)

The author studies fair dice-like hyperbolic systems \(T:M\to M\) equipped with an invariant probability measure \(\mu\). By definition, such systems have a Markov partition \(\alpha= \{a_1,\dotsc ,a_r\}\) such that, for all \(m \in {\mathbb N}\) and \((i_1,\dotsc ,i_m) \in \{1,\dotsc ,r\}^m\), one has \(\mu( \bigcap_{k=1}^{m} T^{-k+1} a_{i_k}) =r^{-m}\). For these systems, the author then considers open dynamical systems \(T_a: M \setminus a \to M\), where \(a\) refers to some cylinder set in the \(m\)-th refinement of \(\alpha\), and studies the survival probability \(P_n(a):= 1- \mu\left( \bigcup_{i=0}^n T^{-i}(a)\right)\). Also, with each of these open systems, he inductively associates the strictly increasing sequence of periods \((p_{a,1}, p_{a,2},\dots)\), where \(p_{a,j}\) refers to the \(j\)-th least period of \(T\)-periodic points in \(a\). The main result of the paper is the following one.NEWLINENEWLINE Let \(a\) and \(b\) be two cylinder sets of the same length, and let \(N\) be given by the conditions \(p_{a,N} <p_{b,N}\) and \(p_{a,k} =p_{b,k}\) for all \(k<N\). Then the associated survival probabilities have the property that \(P_n(a) >P_n(b)\) for all \(n \geq N\).NEWLINENEWLINEVarious examples are considered, among them are the tent map, the von Neumann-Ulam map and baker's map.NEWLINENEWLINEFor the entire collection see [Zbl 1237.37004].