Fast Birkhoff sums in expanding interval maps with infinitely many branches

We investigate expanding maps on the interval $[0,1]$ with the following properties:

- infinitely many branches, image of each covering the whole $[0,1]$ (that is, $(0,1) = \bigcup I_i$ and $f(I_i)=[0,1] \forall i$),

- uniformly estimated distortion (though weaker than bounded distortion),

- polynomially increasing derivative on branches.

The main example we have in mind: the Gauss map.

On such a system we consider a potential $\phi:[0,1]\to (0,\infty)$, constant on each $I_i$ but fast increasing with $i$. Then for a fast increasing gauge function $\Phi:\N\to (0,\infty)$ we define
\[
A(\phi, \Phi) = \{x\in [0,1]; \lim_{n\to\infty} \frac {\sum_{k=0}^{n-1} \phi(f^k(x))} {\Phi(n)} =1\}.
\]
The sets of this type were investigated by Khinchin, Philipp etc. Our goal is to describe $\dim_H A(\phi, \Phi)$ for reasonable choices of $\phi$ and $\Phi$. It is a joint work with Lingmin Liao.