Let $\Omega \subseteq \mathbb{R}$ be a semi-interval split into $d$ disjoint semi-intervals, $\Omega = \bigcup_{j=1}^{d} \Delta_{j}$ , $\Delta_{j}=[\beta_{j-1},\beta_{j})$. An interval translation $T: \Omega \to \Omega$ is a map given by a translation on each $\Delta_{j}$, $T|_{\Delta_{j}}: x \mapsto x+\gamma_{j}$, the vector $(\gamma_{1}, \cdots , \gamma_d)$ is fixed. Here we consider $\Omega$ to be the interval $[0,1)$.
Define $\Omega_{0}= \Omega$, $\Omega_{n}=T\Omega_{n-1}$ for $n \geq 1$, and let $X$ be $\overline{\bigcap\limits_{n=1}^{\infty} \Omega_{n}}$
Let $T$ be any interval translation map. We say that an interval $\Delta \subseteq \Omega$ is $T$-regular if there exists $N \in \mathbb{N}$ such that for any $x \in [0,1)$ there exists $ 1 \leq n < N$ such that $T^{n}(x) \in \Delta$.
Now my question is the following statement from a lemma of a paper written by Denis Volk.
Let $X$ is a cantor set and $\Delta$ be $T$-regular. because $\Delta$ is regular $Y=X \cap \Delta$ is non-empty cantor set.
Is this easy to see that?