Lecture 1: Monday June 14, 11h - Introduction and motivation

The objects of this course will be presented and discussed. (The main example for this first lecture will be the Gauss map on the interval [0,1].) *Transfer operators, their spectrum and its use in (discrete-time) dynamics (physical measures, measures of maximal entropy, equilibrium states, decay of correlations, linear response, limit theorems, application to the distributional analysis of Euclidean algorithms...) *Dynamical zeta functions and determinants, their poles and zeroes and the connection with the spectrum of transfer operators (Ruelle resonances, the Selberg zeta function and quantum chaos, prime number theorems...)

Lecture 2: Tuesday June 15, 11h -Transfer operators and dynamical zeta functions/determinants for expanding and piecewise expanding systems

For smooth expanding systems the results of Ruelle and Gundlach-Latushkin will be evoked, but we will mostly discuss the transfer operator acting on Sobolev spaces. For piecewise smooth system, we will probably restrict to the one-dimensional setting (BV and more recent Sobolev approach). We will emphasize the role of Lasota-Yorke inequalities.

Lecture 3: Wednesday June 16, 11h - Spectral properties of transfer operators for smooth hyperbolic systems

We will present the anisotropic Sobolev spaces introduced by Tsujii and the lecturer, and explain how one can estimate the essential spectral radius of the transfer operator associated to an Anosov or Axiom A diffeomorphism on them. We will first work in a simple model where ordinary "Triebel" spaces can be used, and then sketch the general construction.

Lecture 4: Thursday June 17, 11h : Dynamical determinants for smooth hyperbolic systems

We will relate the zeroes of a dynamical determinant and the spectrum of the transfer operator acting on the Sobolev spaces defined in Lecture 3, using a simplification of the proof published by Tsujii and the author, which is based on approximation numbers and regularised determinants.

Additional information and bibliography at http://www.math.ens.fr/~baladi/porto2010.pdf