Let $X$ be a compact space (e.g. the Cantor Set) and let $T:X\to X$ be a minimal homeomorphism (meaning that every orbit is dense in $X$).
If $U\subseteq X$ is an open subset (clopen if X is the Cantor set), let $T_U:U\to U$ be the first return map i.e., an element $x\in U$ is sent to its first return point in $U$. (Such a point exists by the minimality of $T$). Easy to see that $(U,T_U)$ is another minimal system.
Specific question:
A paper I am reading (link) states (without proof) that if $(X,T)$ is not (conjugate to) an odemeter then so is $(U,T_U)$. I was unable to observe this.
General question:
In general, what relations do the systems $(X,T)$ and $(U,T_U)$ have?
Thank you in advance.