Introduction to Higher Cubical Operads. Second Part: The Functor of Fundamental Cubical Weak $\infty$-Groupoids for Spaces

Some Aspects of Dynamical Topology: Dynamical Compactness and Slovak Spaces

The area of dynamical systems where one investigates
dynamical properties that can be described in topological terms is "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 recently called as "Dynamical
Topology". For (discrete) dynamical systems given by compact metric spaces and
continuous (surjective) self-maps we (mostly) survey some results on 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
omega-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
omega-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 omega-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.

From Euler's play with infinite series to the anomalous magnetic moment