# Does an aperiodic dynamical system have $n$-markers for any $n$?

I was wondering if a certain lemma in an article by Downarowicz holds in a more general setting (see details below):

Let $(X,T)$ be a topological dynamical system. I.e. $X$ is a compact Hausdorff space and $T:X\rightarrow X$ is a continuous mapping.

Let us call an open subset $U\subset X$ an $n$-marker ($n\in\mathbb{N}$) if

1. The sets $T^{-i}(U)$ $(1\leq i\leq n)$ are pairwise disjoint.

2. The sets $T^{-i}(U)$ $(1\leq i\leq m)$ cover $X$ for some $m\geq n$.

The system $(X,T)$ is said to have the marker property if there exist open $n$-markers for all $n\in\mathbb{N}$.

My question is: Does all aperiodic ($T$ doesn't have periodic points) dynamical systems have the marker property?

The above definition is a natural generalization of a definition 2 in

Downarowicz, Tomasz. Minimal models for noninvertible and not uniquely ergodic systems. Israel J. Math. 156 (2006), 93--110.

In particular based on lemma 1 in this article the answer to the above question is yes if one assumes in addition that the system has an aperiodic zero-dimensional factor.

It is easy to see that the answer is yes if $(X,T)$ has a non-trivial minimal (all orbits are dense) factor.

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The definition of an $n$-marker isn't showing up for me; can you fix it? If it's what I think it is, there should be many counterexamples (including an irrational rotation of the circle), but of course this is all contingent upon the definition. – Clinton Conley Nov 1 '11 at 15:01
Thank you Clinton. I fixed it. – Yonatan Gutman Nov 1 '11 at 15:55
I see -- that's a bit different than I thought it would be, so I retract my now-false claim about the irrational rotation. – Clinton Conley Nov 1 '11 at 16:11
Since I thought about a similar problem for a few months myself now, it should be noted (for future readers) that a positive answer was already found by Yonatan Gutman for the case of finite dimensional (invertible) systems. The paper can be found here arxiv.org/abs/1208.5248 – Gabor Szabo Dec 17 '12 at 12:33