SEMINARIO
DI SISTEMI
DINAMICI
ore 16.00
Scuola Normale
Superiore
Pisa
(Sala Conferenze
Collegio Puteano)
Viviane Baladi
CNRS, E.N.S.
Paris
Terrà un
seminario dal titolo:
“Linear
response for generic smooth unimodal maps”
(Joint
with Daniel Smania)
Many
interesting dynamical systems admit a "natural" or "physical" measure, which
describes the asymptotic time averages of a positive Lebesgue measure set of
initial conditions. This measure is called the SRB measure, for
Sinai-Ruelle-Bowen. When a smooth one-parameter family f_t of dynamical systems
admits for all (or many) small paramerers t a unique SRB measure mu_t, it is
natural to ask if the map t -> mu_t is also smooth(possibly in the sense of
Whitney). In 1997, David Ruelle solved the case when the f_t are smooth and
uniformly hyperbolic, obtaining a formula for the derivative: the linear
response formula.Ten years later, Daniel Smania and I discovered by studying a
"toy model" (piecewise expanding maps) that the presence of critical points
(which destroy structural stability) may cause obstructions to linear response.
We proved that the condition of "tangency" of the family to the topological
class was necessary and sufficient for linear response to hold. For smooth
nonuniformly hyperbolic unimodal maps, the situation is much more difficult, and
for the moment one only considers families f_t which remain in the topological
class of f_0. Up to 2009, only (nongeneric) situations very close to hyperbolic
had been tackled (Ruelle, B-Smania). We shall present a recent joint result with
Daniel Smania (available on arxiv.org), which holds under a generic recurrence
condition.
Tutti gli
interessati sono invitati a partecipare.
Classe
di Scienze