Breakdown of linear response for smooth families of dynamical systems with bifurcations

School of Mathematics Colloquium
Thursday, October 24, 2013 - 11:00am
1 hour (actually 50 minutes)
Skyles 006
Ecole Normale Superieure, Paris
 (Joint with: M. Benedicks and D. Schnellmann) Many interesting dynamical systems possess a unique SRB ("physical")measure, which behaves well with respect to Lebesgue measure. Given a smooth one-parameter family of dynamical systems f_t, is natural to ask whether the SRB measure depends smoothly on the parameter t. If the f_t are smooth hyperbolic diffeomorphisms (which are structurally stable), the SRB measure depends differentiably on the parameter t, and its derivative is given by a "linear response" formula (Ruelle, 1997). When bifurcations are present and structural stability does not hold, linear response may break down. This was first observed for piecewise expanding interval maps, where linear response holds for tangential families, but where a modulus of continuity t log t may be attained for transversal families (Baladi-Smania, 2008). The case of smooth unimodal maps is much more delicate. Ruelle (Misiurewicz case, 2009) and Baladi-Smania (slow recurrence case, 2012) obtained linear response for fully tangential families (confined within a topological class). The talk will be nontechnical and most of it will be devoted to motivation and history.  We also aim to present our new results on the transversal smooth unimodal case  (including the quadratic family), where we obtain Holder upper and lower  bounds (in the sense of Whitney, along suitable classes of parameters).