# Regularity of the pdf of partial Birkhoff sums

Suppose that $$T: X \to X$$ is some measurable map on a Riemannian manifold $$X$$ (possibly with boundary). Let $$\mu$$ denote the Riemannian measure on $$X$$. For measurable, real-valued $$g$$ we may consider the partial Birkhoff sums $$S_n := \sum_{k=0}^{n-1} g \circ T^k$$. I am specifically interested in the case where $$T$$ possesses some hyperbolicity e.g. $$T$$ could be a piecewise $$\mathcal{C}^2$$ expanding map on $$[0,1]$$, or a $$\mathcal{C}^2$$ Anosov map on $$\mathbb{T}^2$$. Also, $$g$$ may be as regular as needed.

If $$g$$ and $$T$$ have enough regularity, then $$S_n$$ has a probability density function $$f_n$$. My question is whether anybody has studied the regularity of $$f_n$$: under what conditions does $$f_n'$$ exist, and what can we say about it as $$n \to \infty$$. If we define the Fischer information of $$S_n$$ to be $$I(S_n) = \int_{\mathbb{R}} \left[\frac{\mathrm{d}}{\mathrm{d}x} \log f_n\right]^2 f_n(x) \, \mathrm{d}x,$$ can we say whether $$I(S_n)$$ is finite? If so, can we say anything about the growth of $$I(S_n)$$ as $$n \to \infty$$?