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 \(\varphi :T\times X\rightarrow X\) $\phi :T\times X\to X$ , or simply (T, X), be any dynamical system on a space X with a topological semigroup T. We say that \(x\in X\) $x\in X$ 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 \(d_{F\phi }(N(x, U)>0\) $d_{F\varphi }(N(x,U)>0$ , respectively, where U is a neighborhood of the point x and \(N(x, U)=\{t: tx\in U\}\) $N(x,U)=\{t:tx\in U\}$ . 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 \(f:X\rightarrow \mathbb {R_+}\) $f:X\to {\mathbb{R}}_{+}$ is upper semicontinuous and show that \(x\in X\) $x\in X$ is uniformly recurrent if and only if it is an f-uniformly recurrent point for every upper semicontinuous \(f:X\rightarrow \mathbb {R_+}\) $f:X\to {\mathbb{R}}_{+}$ on the regular space X. In the case of metric space X, \(f:X\rightarrow \mathbb {R}_+\) $f:X\to {\mathbb{R}}_{+}$ is a continuous function. Also, \(x\in X\) $x\in X$ is a uniformly recurrent point if and only if \(x\in \overline{Ax}\) $x\in \overline{\mathit{Ax}}$ 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.