Abstract

If f_t is a smooth parametrised family of dynamical systems admitting a unique SRB measure (for all, or for many, values of the parameter t), it is natural to ask whether the SRB measure depends smoothly on the dynamics. In dimension one, SRB measures are absolutely continuous invariant measures with a positive exponent. With Daniel Smania, we showed that the SRB measure is differentiable at 0 if and only if the path f_t is tangent to the topological class of f_0 horizontality), for piecewise expanding unimodal maps. In that case, we recover a resummation of Ruelle′s divergent candidate for the value of the derivative of the SRB measure.
We will present this result and explain why our smooth deformations theory shows that horizontality is a codimension one condition.
Then, we shall move to analytic unimodal maps and briefly present Ruelle′s result for Misiurewicz maps and our new result on Collet-Eckmann maps. If time permits, we shall end by some conjectures in the differentiable Collet-Eckmann setting and open questions in higher dimensions.

Joint work with D. Smania.