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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm123-2-4,
author = {Tomasz Downarowicz},
title = {Two commuting maps without common minimal points},
journal = {Colloquium Mathematicae},
volume = {122},
year = {2011},
pages = {205-209},
zbl = {1228.37011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm123-2-4}
}
Tomasz Downarowicz. Two commuting maps without common minimal points. Colloquium Mathematicae, Tome 122 (2011) pp. 205-209. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm123-2-4/