Assume that $ T: X to X $ is a measurable map on a Riemannian variety $ X $ (possibly with limit). Let $ mu $ denote the Riemannian measure on $ X $. For measurable, real value $ g $ we can consider the partial sums of Birkhoff $ S_n: = sum_ {k = 0} ^ {n-1} g circ T ^ k $.

I am particularly interested in the case where $ T $ has some hyperbolicity, for example $ T $ could be a piece $ mathcal {C} ^ 2 $ map expanding on $[0,1]$, or one $ mathcal {C} ^ 2 $ Anosov map on $ mathbb {T} ^ 2 $. Also, $ g $ can be as regular as necessary.

Yes $ g $ and $ T $ to have enough regularity, then $ S_n $ has a probability density function $ f_n $. My question is whether anyone has studied the regularity of $ f_n $: in what conditions $ f_n $ exists, and what can we say about it $ n to infty $. If we define the Fischer information of $ S_n $ be

$$

I (S_n) = int _ { mathbb {R}} left[frac{mathrm{d}}{mathrm{d}x} log f_nright]^ 2 f_n (x) , mathrm {d} x,

$$

can we say if $ I (S_n) $ is finished? If so, can we say something about the growth of $ I (S_n) $ as $ n to infty $?