Ergodic Theory and Dynamical Systems

On the topological entropy of saturated sets

C.-E. PFISTER a1 and W. G. SULLIVAN a2
a1 École polytechnique fédérale de Lausanne, Institut d'analyse et calcul scientifique, CH-1015 Lausanne, Switzerland (e-mail:
a2 School of Mathematical Sciences, UCD Dublin, Belfield, Dublin 4, Ireland (e-mail:

Article author query
pfister ce   [Google Scholar] 
sullivan wg   [Google Scholar] 


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).

(Published Online February 12 2007)
(Received November 19 2005)
(Revised September 1 2006)