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 (for which we have recently found a new
proof, more suitable to higher-dimensional generalisations)
and explain why our smooth deformations theory shows that
horizontality is a codimension one condition.
Then, we shall move to analytic unimodal maps and present
Ruelle's result for Misiurewicz maps and our result on
Collet-Eckmann maps. We shall end by some conjectures
in the differentiable Collet-Eckmann setting and open questions in higher
dimensions.