Speaker
Prof.
Daniele Galli
(University of Zurich, Switzerland)
Description
We study the skew product systems $T:\mathbb{S}^1\times\mathbb{R}^d\to\mathbb{S}^1\times\mathbb{R}^d$, $T(x,y)=(\ell x, A y+\phi(x)),$ where $\ell\geq 2,$ $A \in GL_d(\mathbb{R})$, with $\rho(A)<1$, and $\phi \in C^r(\mathbb{S}^1, \mathbb{R}^d)$. We show that for a generic $\phi$:
if $|\det(A)|\ell<1$, the Hausdorff dimension of the solenoid attractor and the SRB measure dimension equals the affinity dimension;
if $|\det(A)|\ell>1$, the SRB measure is absolutely continuous.
In the latter case we investigate the Sobolev regularity of the density of the SRB measure. We introduce a derivative dispersion condition, which replaces the usual transversality condition.
This is a joint work with E.Passaglia and A.Ulliana