Two commuting maps without common minimal points
Tom 123 / 2011
Colloquium Mathematicum 123 (2011), 205-209 MSC: 37B20, 37B05. DOI: 10.4064/cm123-2-4
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.