1
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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? $\endgroup$ – Alex Sep 4 '14 at 18:38
  • $\begingroup$ 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. $\endgroup$ – MHS Sep 5 '14 at 1:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.