Maybe we can consider a large digit instead of $\{0\,,1\}$

Thanks, I will think your idea carefully and will find such kind of sequences

I will consider your comment carefully, Thanks a lot.

My question is that if the length of $(a\,,b)$ is small, can we say something about $\Omega$, we should note the importance of the assumption.Sidorov prove that we only need to consider $(a\,,b)\subset(0\,,\dfrac{1}{2})$ or $(a\,,b)\in (\dfrac{1}{4}\,,\dfrac{1}{2})\times(\dfrac{1}{2}\,,\dfrac{3}{4})$ in his paper: The doubling map with asymmetrical holes.

Yes. I know this paper:(Topological and symbolic dynamics for hyperbolic systems with holes. ), in general. this lemma is false, because we can consider the hole is very big,and the only orbit can escape the hole is 0, for example $(a\,,b)=(0.1\,,0.8)$

Thanks very much. Actually, I was reading your paper, and I want to study some simple and further problems which depend on your recent paper.

We can consider $b-a<0.1$ as we can see, if the interval is bigger, then $\Omega$ may be just one pont, generally, if the map T is defined as a continuous map, this lemma is not true, therefore, we can consider the discontinuous points.