4,906 research outputs found

    Dynamical consequences of a free interval: minimality, transitivity, mixing and topological entropy

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    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

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    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 DD_\infty is more characteristic than the others because it is the unique one with a dense conjugacy class. For this group GG_\infty, 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 GG_\infty. This allows us to prove that point-stabilizers in GG_\infty are amenable and to describe the universal Furstenberg boundary of GG_\infty.Comment: Slight modifications about the expositio
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