I am trying to prove an analytic result for gesodesic flows on negatively curved manifolds and I encountered the following dynamical-system porblem.


Let $B^n$ be $n$-dimensional balls and $h:B^{n-k}\times B^{k} \to U$ a $k$-dimensional foliation chart with $C^\infty$ leaves. i.e. For $x\in B^{n-k}$, $W\_x := h(\{x\}\times B^k)\subset U$ are $C^\infty$ submanifolds. Furthermore let $y\in B^{n-k}$ such that $L=h(\{y\}\times B^{n-k})$ is a smooth transversal. Then according to Brin-Stuck "Introduction to Dynamical Systems" the foliation is *absolutely continuous* if there is a measurable family of positive measurable functions $\delta\_x: W\_x\to\mathbb R$ such that for all measurable sets $A\subset U$
$$\int\_U \mathbf 1\_A(x,y) dm\_{euc} = \int\_L \int\_{W\_x} \mathbf 1\_A(x,y) \delta\_x(y) dm\_{W\_x}(y)dm\_L(x).$$
Here $dm\_{euc}$ is the Eucildean measure on $U$ and $dm\_{W\_x}$ and $dm\_L$ the measures obtained by restricting $dm\_{euc}$ on the smooth submanifolds.


If the foliation ist the stable foliation of an Anosov differomorphism then I found that it is absolutely continuous. It think it is clear that one cannot hope that $\delta\_x(y)$ depends $C^\infty$ on $x$ unless the foliation is smooth. 


**My question is**: Under which conditions on the foliation can one obtain, that for any $x$ one has $\delta\_x \in C^\infty(W\_x)$(ideally I would also like to have, that $\delta\_x$ depends continuousely on $x$ w.r.t. to the $C^\infty$-topology, but maybe I can also handle the case if it is only measureable)? And more precisely: Is such a statement always true for the strong stable foliation of a $C^\infty$ Anosov flow. If it doesn't hold for general Anosov Flows: What are the weakest known assumption that it holds? (If I understand it correctly then $C^1$ foliations implies this statement via Fubinis theorem). 



Smooth conditional measures for strong stable foliations of Anosov flows
Due to helpfull discussions with several dynamical systems experts I am now able to give an answer to the question, that I asked about one year ago and provide some references:


The short answer is, that for the smoothness of the conditional measures it is sufficiant to know the smoothness of the holonomy maps with respect to a smooth transversal foliation. A detailed explanation of this argument can for example be found in Appendix A of [arxiv-version](https://arxiv.org/abs/1511.08338) but the techniques are rather standard, see e.g. Brin-Stuck "Introduction to dynamical systems" Section 6.2.


Concerning the stable and unstable foliations of an $C^\infty$-Anosov flow, a statement on the smoothness of the holonomy maps can for example found in [Giulietti-Liverani-Pollicott Ann.Math (2013)](http://annals.math.princeton.edu/2013/178-2/p06) (Appendix E). Thus one obtains the smoothness of the conditional densities for $C^\infty$-Anosov flows. A precise formulation of this statement is now given in Theorem 7 [arxiv-version](https://arxiv.org/abs/1511.08338). 


If one is interested in an analogouse statement for Anosov map, there is a remark in [Chernov "Invariant measures for hyperbolic dynamical systems" (Handbook of Dynamical systems 1A)](https://zbmath.org/?q=an:02079132), stating the smoothness of the conditional densities and giving a rough idea how to prove it.


