# Survivor sets for expanding maps of the interval

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 $$\dim_H \mathcal J(I_n)\to 1, \quad n\to \infty?$$

• I am not sure that this matters much, but I guess you mean either "piecewise smooth" or you want to replace the interval with the circle (otherwise, there is no such map). Apr 20, 2016 at 20:01
• Also, I have a problem with variables: there are a $n$ outside the set you are defining, and a quantified $n$ inside it, so I guess there should be two different numbers and it is not clear what you mean. Apr 20, 2016 at 20:05

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 $k \to \infty$.