4,906 research outputs found
Dynamical consequences of a free interval: minimality, transitivity, mixing and topological entropy
We study dynamics of continuous maps on compact metrizable spaces containing
a free interval (i.e., an open subset homeomorphic to an open interval). A
special attention is paid to relationships between topological transitivity,
weak and strong topological mixing, dense periodicity and topological entropy
as well as to the topological structure of minimal sets. In particular, a
trichotomy for minimal sets and a dichotomy for transitive maps are proved.Comment: 21 page
Topological properties of Wazewski dendrite groups
Homeomorphism groups of generalized Wa\.zewski dendrites act on the infinite
countable set of branch points of the dendrite and thus have a nice Polish
topology. In this paper, we study them in the light of this Polish topology.
The group of the universal Wa\.zewski dendrite is more
characteristic than the others because it is the unique one with a dense
conjugacy class. For this group , we show some of its topological
properties like existence of a comeager conjugacy class, the Steinhaus
property, automatic continuity and the small index subgroup property. Moreover,
we identify the universal minimal flow of . This allows us to prove
that point-stabilizers in are amenable and to describe the universal
Furstenberg boundary of .Comment: Slight modifications about the expositio
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