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

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

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.