Homeomorphisms with Markov partitions
Markov partitions were first constructed for Anosov diffeomorphisms by \textit{Ya. G. Sinai} [Funct. Anal. Appl. 2, 245-253 (1968); translation from Funkts. Anal. Prilozh. 2, No.3, 70-80 (1968; Zbl 0194.22602)]. After that, \textit{R. Bowen} [Am. J. Math. 92, 725-747 (1970; Zbl 0208.25901)] showed the existence of Markov partition on nonwandering sets of Axiom A diffeomorphisms. In these papers the notion of canonical coordinate plays an important role to construct Markov partitions. \textit{K. Hiraide} [Tokyo J. Math. 8, 219-229 (1985; Zbl 0639.54030)], in purely topological setting, proved the existence of Markov partition for expansive homeomorphisms with pseudo orbit tracing property (POTP) by constructing canonical coordinates. For example, every expansive automorphism of a solenoidal group has POTP, and hence a canonical coordinate as well as a Markov partition. However homeomorphisms with Markov partitions do not necessarily have canonical coordinates. In fact, every pseudo-Anosov map has a Markov partition and does not have canonical coordinates (see paragraphs 9 and 10 of \textit{A. Fathi} and \textit{F. Laudenbach}, Astérisque 66-67, 139-150 (1979; Zbl 0446.57015)). Thus it is natural to ask what kind of expansive homeomorphisms have Markov partitions. The purpose of this paper is to give necessary and sufficient conditions for expansive homeomorphisms to have Markov partitions.
scientific article; zbMATH DE number 4169718