Mini-courses:
A. HAMMERLINDL / R. POTRIE (6 hours, 1st and 2nd week)
Classification of partially hyperbolic diffeomorphisms in certain 3-manifolds
Abstract: We will present our recent work on the classification of partially hyperbolic diffeomorphisms on 3-dimensional manifolds with solvable fundamental group. This family includes, in particular, all torus bundles over the circle and includes manifolds which support Anosov flows. These systems can be related to simple algebraic examples by means of a notion of "leaf conjugacy" as introduced by Hirsch, Pugh, and Shub. The course will also present previous results and examples, including results on the existence and uniqueness of foliations invariant under the dynamics.
J. BOCHI / C. BONATTI (6 hours, 1st week)
C1- perturbation techniques in the neighbourhood of periodic orbits
Abstract: The C1- topology allows to perform local perturbations in arbitrarily small neighbourhood of a finite set, and we are specially interested in perturbation in the neighbourhood of periodic orbits. These perturbations can be seen as perturbations of the linear cocycle defined by the differential along the periodic orbit. Many results give answers to the following questions:
- what are the Lyapunov exponents we can obtain by perturbations of the linear cocycle of the differential along a periodic orbit of very long period?
- what control do we have on the invariant (stable and unstable) manifolds of the periodic orbits under these perturbations?
We will see that the main restriction comes from the dominated splitting carried along the periodic orbits. We will try to present a survey of results and arguments, starting with an argument of Mañé and arriving to very recent works.
S. GAN (4 hours, 2nd week)
Partially hyperbolic singular flows
Abstract: I will talk about two methods in dealing with the difficulty induced by singularity of flow. First, I will introduce the concept of extended linear Poincare flow and then prove that domination of linear Poincare flow implies the Lorenz-like property of singularity. Then, I will explain the uniformity of scaled Poincare flow and introduce the shadowing of quasi-hyperbolic pseudo-orbit, and then give some applications in the study of singular star flows and density conjectures for flows.
C. LIVERANI (4 hours, 1st week)
Fast-slow systems: beyond averaging and stable ergodicity.
Abstract: I will briefly explain how some important non-equilibrium statistical mechanics problems can be reduced to the study of the statistical properties of special partially hyperbolic systems: small perturbations of a trivial extension. This are examples of slow-fast systems but averaging theory is far from providing the needed results, nor does stable ergodicity help much. I will try to explain the state of the (quite open) problem and discuss its relation to various limit theorems.
K. GELFERT / M. RAMS (6 hours, 1st and 2nd weeks)
Dimension and Lyapunov exponents in conformal non-hyperbolic dynamics
Abstract: We study the fractal properties of certain level sets for a class of conformal maps that includes several non-hyperbolic situations. We focus on Hausdorff and box counting dimension and analyze level sets of Lyapunov exponents and local dimensions. We present some methods that apply in several - similar - contexts such as parabolic Markov maps of the interval, rational maps of the Riemann sphere, and certain conformal flows. We indicate (tentatively) the following lectures:
1. Multifractal spectrum for conformal repellers
2. Nonuniformly hyperbolic systems (expansive Markov maps of the interval)
3. Nonuniformly hyperbolic systems (analysis of weak Gibbs measures)
4.-5. Extension of methods to rational maps and flows
6. Expanding interval maps with infinitely many branches
A. KATOK (4 hours, 2nd week)
Measure rigidity beyond uniform hyperbolicity: an introduction
The non-uniform measure rigidity program that has been pursued by Boris Kalinin, Federico Rodriguez Hertz and the speaker since 2006 aims at showingthat actions of higher rank abelian groups that are hyperbolic in a weak sense ( i.e. possessing an ergodic invariant measure with non-zero Lyapunov exponents satisfying some entropy and non-degeneracy conditions) are rigid in various sense. Sometimes existence of such a measure can be deduced from global homotopy or homology data. Precise meaning of relevant notions of rigidity will be explained in the course. This program is based on three advanced techniques: The theory of non-uniformly hyperbolic systems (aka Pesin theory), measure rigidity as it was developed for algebraic actions, and the theory of non-stationary non-uniform normal forms. While the basics of the first of those will be assumed to be known to the audience, the other two will be introduced and briefly surveyed. After that some key ideas used in the non-uniform measure rigidity will be explained and principal results outlined.