Two commuting maps without common minimal points

Tomasz Downarowicz. "Two commuting maps without common minimal points." Colloquium Mathematicae 123.2 (2011): 205-209. <http://eudml.org/doc/284111>.

@article{TomaszDownarowicz2011,
abstract = {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.},
author = {Tomasz Downarowicz},
journal = {Colloquium Mathematicae},
keywords = {topological dynamical system; commuting homeomorphisms; minimal set; minimal point},
language = {eng},
number = {2},
pages = {205-209},
title = {Two commuting maps without common minimal points},
url = {http://eudml.org/doc/284111},
volume = {123},
year = {2011},
}

TY - JOUR
AU - Tomasz Downarowicz
TI - Two commuting maps without common minimal points
JO - Colloquium Mathematicae
PY - 2011
VL - 123
IS - 2
SP - 205
EP - 209
AB - 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.
LA - eng
KW - topological dynamical system; commuting homeomorphisms; minimal set; minimal point
UR - http://eudml.org/doc/284111
ER -