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 $n \to \infty$?


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