Course on anisotropic Banach spaces and dynamics 3
Advanced &Ph. D course
"A tour in the jungle of anisotropic Banach spaces for hyperbolic dynamics"
Ruelle transfer operators are a key tool to study the statistical properties of dynamical systems with enough hyperbolicity and smoothness. We consider (mostly discrete-time)
C^r (or piecewise C^r) dynamics, for r>1. We shall start with the simplest case of smooth expanding dynamics, where the relevant spectrum is obtained by letting the operator act on a Banach or Hilbert space of smooth functions.
We shall then move to hyperbolic dynamics, for which we shall describe several of the available anisotropic spaces used to study the spectra of weighted transfer operators. Topics should include stability of the spectrum and statistical properties of Gibbs states - such as the SRB measure or the measure of maximal entropy.
If time permits, we shall discuss the dynamical determinants (or zeta functions).