Locally maximal attractors of expanding dynamical systems

Abstract

UDC 517.9

We study locally maximal attractors of expanding dynamical systems.  Our main result is a representation of these attractors with the help of   topological Markov chains corresponding to the Markov partitions of these attractors, which allows to describe the system dynamics on them. 

Ya. G. Sinai was the first who constructed and used Markov partitions for Anosov's diffeomorphisms [Funk.  Anal. Prilozh., 2, No. 1, 64–89; No. 3, 70–80 (1968); English translation:  Funct. Anal.  Appl., 2, No. 1, 61–82; No. 3, 245–253 (1968)].  Expanding endomorphisms regarded as  the simplest representatives of endomorphisms, were first studied by M. Shub [Amer. J. Math., 91, No. 1, 175–200 (1969)].  To construct Markov partitions for expanding endomorphisms, we modernize Sinai's method in a corresponding way. 

A more detailed  historical overview can be found in the work by O. M. Sharkovsky [Ukr. Mat. Zh., 74, No. 12, 1709–1718 (2023);  English translation:  Ukr. Math.  J., 74, No. 12, 1950–1960 (2023)]. There Sharkovsky indicated that the methods used to prove the main results [Dokl. Akad. Nauk SSSR, 170, No. 6, 1276–1278 (1966);  English translation:  Soviet Math. Dokl., 7, No. 5, 1384–1386 (1966)] were, in fact, published in the  collection of papers  ``Dynamical systems and the problems of stability of solutions of differential equations'' (1973) issued by the Institute of Mathematics of the Academy of Sciences of Ukraine.  This collection is difficult to access and it was never translated into English. However, in the cited paper these methods were applied to somewhat different objects.  To the best of the  authors' knowledge, there is no information about publications of similar results.  Given the outlined history and importance of this approach (based on the Markov partitions and topological Markov chains) for the description of  attractors' construction it seems reasonable  to rеpublish these results  in a new way.