Gene

Metrization theorem

Homeomorphism (graph theory)

Combinatorics

Discrete mathematics

Uniform continuity

Mathematical analysis

Topological manifold

Topological tensor product

Topology (electrical circuits)

Chemistry

Compact space

Transitive relation

Hausdorff space

Hausdorff dimension

Metric space

Pure mathematics

Connected space

Mathematics

Topological dynamics

Topological vector space

Functional analysis

Biochemistry

Limit point

Separable space

Hausdorff measure

Topological space

Urysohn and completely Hausdorff spaces

Discrete and Continuous Dynamical Systems

We consider various notions from the theory of dynamical systems from a topological point of view. Many of these notions can be sensibly defined either in terms of (finite) open covers or uniformities. These Hausdorff or uniform versions coincide in compact Hausdorff spaces and are equivalent to the standard definition stated in terms of a metric in compact metric spaces.We show for example that in a Tychonoff space, transitivity and dense periodic points imply (uniform) sensitivity to initial conditions. We generalise Bryant's result that a compact Hausdorff space admitting a $c$-expansive homeomorphism in the obvious uniform sense is metrizable. We study versions of shadowing, generalising a number of well-known results to the topological setting, and internal chain transitivity, showing for example that $ω$-limit sets are (uniform) internally chain transitive and weak incompressibility is equivalent to (uniform) internal chain transitivity in compact spaces.

What is topological about topological dynamics?