Two commuting maps without common minimal points

Tomasz Downarowicz¹

¹ Institute of Mathematics and Computer Science Wrocław University of Technology Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland

Colloquium Mathematicum, Tome 123 (2011) no. 2, pp. 205-209.

Résumé

We construct an example of two commuting homeomorphisms $S$, $T$ of a compact metric space $X$ such that the union of all minimal sets for $S$ is disjoint from the union of all minimal sets for $T$. In other words, there are no common minimal points. This answers negatively a question posed in [C-L]. We remark that Furstenberg proved the existence of “doubly recurrent” points (see [F]). Not only are these points recurrent under both $S$ and $T$, but they recur along the same sequence of powers. Our example shows that nothing similar holds if recurrence is replaced by the stronger notion of uniform recurrence.

Détail

DOI : 10.4064/cm123-2-4

Keywords: construct example commuting homeomorphisms compact metric space union minimal sets disjoint union minimal sets other words there common minimal points answers negatively question posed c l remark furstenberg proved existence doubly recurrent points see only these points recurrent under recur along sequence powers example shows nothing similar holds recurrence replaced stronger notion uniform recurrence