Dynamical topology: Slovak spaces and dynamical compactness

Sergiy Kolyada
Institute of Mathematics National Academy of Sciences of Ukraine, Kiev

Date(s) : 12/12/2017   iCal
11 h 00 min - 12 h 00 min

The area of dynamical systems where one investigates dynamical properties that can be described in topological terms is called « Topological Dynamics ». Investigating the topological properties of spaces and maps that can be described in dynamical terms is in a sense the opposite idea. This area is called « Dynamical Topology ».

For (discrete) dynamical systems given by compact metric spaces and continuous (surjective) self-maps, I will mostly be talking about two new notions: « Slovak Space » and « Dynamical Compactness ». Slovak Space is a dynamical analogue of the rigid space: a nontrivial compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism.

Dynamical Compactness is a new concept of chaotic dynamics. The ω-limit set of a point is a basic notion in theory of dynamical systems and means the collection of states which « attract » this point while going forward in time. It is always nonempty when the phase space is compact. By changing the time we introduced the notion of the ω-limit set of a point with respect to a Furstenberg family. A dynamical system is called dynamically compact (with respect to a Furstenberg family) if for any point of the phase space this ω-limit set is nonempty. A nice property of dynamical compactness:
all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property.

Based on a work by Tomasz Downarowicz, Lubomir Snoha and Dariusz Tywoniuk, and joint works with Wen Huang, Danylo Khilko, Alfred Peris, Julia Semikina and Guohua Zhang.



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