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

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    $\begingroup$ 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). $\endgroup$ Apr 20, 2016 at 20:01
  • $\begingroup$ 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. $\endgroup$ Apr 20, 2016 at 20:05

1 Answer 1


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.


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