Revisiting variations in topological transitivity

Topological dynamical systems (X, T) are actions $$T\,{ \times }\, X \rightarrow X$$ , given as $$(t, x) \mapsto tx$$ on a compact Hausdorff topological space with T an acting group or monoid. We consider the property of transitivity especially for semiflows S) and discuss variations in its definitions. emphasize those properties that differ compared to flows (semi)cascades.