# Dynamical Systems Question on Definition

I'm working through some old notes on Dynamical systems, and I see a definition that I'm not familiar with. I'll call it property B, because I'm not sure what else to call it. In the notes, we assume we are in a compact metric space $X$.

A dynamical system $T: X \rightarrow X$ has property B if for every non-empty open set $U \subseteq X$, there exists some $n \geq 0$ such that $T^n(U) = X$.

Has anyone ever seen this or used this before? It's evidently stronger than transitivity and mixing, but not minimality.

I seem to have seen this before in a class I took in Ergodic Theory. If I remember right it was called Topologically Exact. But I don't see much when I type that as a google search.

• Looking at Katok's book, this would be an equivalent definition to transitivity, if you were in a separable compact metric space. Did you mean to also include separable in your assumption? – Alex Sep 4 '14 at 18:38
• To elaborate a bit more on what Alex just said: In a separable compact metric space existence of a dense orbit (i.e. transitivity) implies the apparently stronger property of having a dense G$_\delta$ set of points with dense orbit. Then taking $U$ non-empty and open implies that $U$ contains points with dense orbit and thus your property B holds in that case. The proof is using Baire's theorem and can be found in several books on topological dynamical systems. – MHS Sep 5 '14 at 1:19