TITLE: Survivor sets for expanding maps of the interval
QUESTION [5 upvotes]: 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 
$$
\dim_H \mathcal J(I_n)\to 1, \quad n\to \infty?
$$

REPLY [2 votes]: 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.