Let $T:[0,1]\to [0,1]$ be a piecewise smooth expanding map, i.e., $|T'(x)|>1$ for all $x$. Let $I\_n$ be a sequence of nested intervals (i.e., $I\_{n+1}\subset I\_n$) such that the length of $I\_n$ tends to 0 as $n\to\infty$. 


Define the *survivor set* for $I\_n$ as follows:
$$
\mathcal J(I\_n)=\{x \in[0,1] : T^k(x)\notin I\_n, \forall k\ge0\}.
$$
**Question.** Is it true that 
$$
\dim\_H \mathcal J(I\_n)\to 1, \quad n\to \infty?
$$



Survivor sets for expanding maps of the interval
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.


