Weighted entropy formulae on Feldman-Katok metric (Q6568943)

Let \(X\) be a compact metric space with a metric \(d\), and let \(T : X \to X\) be a continuous map. A topological dynamical system (TDS) is represented by the triplet \((X, T, d)\), or simply \((X, T)\) when the metric \(d\) is clear from the context. We use \(M(X)\), \(M(X, T)\), and \(E(X, T)\) to denote the sets of all Borel probability measures on \(X\), the set of \(T\)-invariant Borel probability measures on \(X\), and the set of ergodic Borel probability measures on \(X\), respectively.\N\NIn this paper, the authors introduce the notions of weighted Feldman-Katok (FK) metric and weighted FK topological entropy. The authors demonstrate that the weighted topological entropy defined using the FK is equivalent to the weighted topological entropy defined by the Bowen metric. They derive both a Brin-Katok formula and a Katok formula for the weighted FK metric.\N\NLet \((X_1, T_1)\) and \((X_2, T_2)\) be topological dynamical systems, and let \(\pi: X_1 \to X_2\) be a factor map. The authors firstly prove that the \(\textbf{a}\)-weighted topological entropy and the \(\textbf{a}\)-weighted FK topological entropy of \(T_1\) on \(X_1\) are equal, i.e.,\N\[\Nh_{\text{top}}^{\textbf{a}}(X_1, T_1) = h_{\text{FK}}^{\textbf{a}}(X_1, T_1),\N\]\Nwhere \(h_{\text{top}}^{\textbf{a}}(X_1, T_1)\) and \(h_{\text{FK}}^{\textbf{a}}(X_1, T_1)\) represent the \(a\)-weighted topological entropy and the \(\textbf{a}\)-weighted FK topological entropy of \(T_1\) with respect to \(X_1\), respectively.\N\NAdditionally, the authors establish that the weighted measure-theoretic entropy formulas remain valid for the FK metric.