On the topological entropy of saturated sets

Abstract

Let $(X,d,T)$ be a dynamical system, where $(X,d)$ is a compact metric space and $T:X\rightarrow X$ a continuous map. We introduce two conditions for the set of orbits, called respectively the $\texttt{g}$-almost product property and the uniform separation property. The $\texttt{g}$-almost product property holds for dynamical systems with the specification property, but also for many others. For example, all $\beta$-shifts have the $\texttt{g}$-almost product property. The uniform separation property is true for expansive and more generally asymptotically $h$-expansive maps. Under these two conditions we compute the topological entropy of saturated sets. If the uniform separation condition does not hold, then we can compute the topological entropy of the set of generic points, and show that for any invariant probability measure $\mu$, the (metric) entropy of $\mu$ is equal to the topological entropy of generic points of $\mu$. We give an application of these results to multi-fractal analysis and compare our results with those of Takens and Verbitskiy (Ergod. Th. & Dynam. Sys.23 (2003), 317–348).