The answer of your question is yes if $T \in C^2$. It holds since $\dim_H(\mathcal{J}(a,b))$ varies continuously when the end points of the removed intervals varies continuously. Then, by continuity, the Hausdorff dimension of $\mathcal{J}(I_k)$ must tend to $1$ as $k \to \infty$.

This is proved by Urbanski in two old-ish papers:

- Hausdorff Dimension of Invariant Sets for Expanding Maps of a Circle.
*Ergod. Th. and Dynam. Sys.* 6 (1986), 295-309.
- Invariant Subsets of Expanding Mappings of the Circle.
*Ergod. Th. and Dynam. Sys.* 7 (1987), 627-645.

It is worth mentioning that the result also holds if a finite number of intervals are removed.

There is also a relationship between the Hausdorff dimension and the escape rate function. This is in a paper by Keller and Liverani: Rare events, e.scape rates and quasistationarity: some exact formulae. *Journal of Statistical Physics 135 (3)*, 519-534.

I hope this helps.