Various Recurrence and Topologically Sensitive for Semiflows.

We recall various notions of size for a topological semigroup T, not necessarily discrete, such as GH-syndetic set, syndetic set, and positive Følner density set. We give new information about these sets and give some examples to study the relation between them. Let φ : T × X → X , or simply (T, X), be any dynamical system on a space X with a topological semigroup T. We say that x ∈ X is a uniformly recurrent point, almost periodic point of von Neumann, or weakly uniformly recurrent point, if the return time set N(x, U) is syndetic, GH-syndetic, or d F ϕ (N (x , U) > 0 , respectively, where U is a neighborhood of the point x and N (x , U) = { t : t x ∈ U } . It is known that there is no relation between the set of uniformly recurrent points and the set of almost periodic points of von Neumann for a semiflow (T, X). We give further examples for it. We introduce a notion of f-uniformly recurrent point, where f : X → R + is upper semicontinuous and show that x ∈ X is uniformly recurrent if and only if it is an f-uniformly recurrent point for every upper semicontinuous f : X → R + on the regular space X. In the case of metric space X, f : X → R + is a continuous function. Also, x ∈ X is a uniformly recurrent point if and only if x ∈ Ax ¯ for every thick set A of T. Assume that S is a closed normal non-trivial subsemigroup of T. We prove that every uniformly recurrent point of (S, X) is a uniformly recurrent point of (T, X). The converse holds if T is a discrete semigroup. Let (T, X) be a semiflow on topological space X. Then we show that every two nonempty open sets in X share an orbit of a weakly uniformly recurrent point of (T, X) if and only if (T, X) is a topologically transitive with a dense set of weakly uniformly recurrent points. Finally, we give topological version of sensitive dependence on the initial condition for semiflow (T, X) on topological space X and we show that if the semiflow (T, X) is nonminimal and every two non-empty open sets share an orbit of a weakly uniformly recurrent point, then (T, X) is syndetic-sensitive. [ABSTRACT FROM AUTHOR]