Dependence of SRB measures on chaotic dynamics

SRB measures are invariant probability measures associated to many interesting chaotic dynamics, and which often have strong statistical features (exponential decay of correlations, limit theorems...).
In this course, we shall consider smooth families of (mostly discrete time) chaotic dynamical systems f_t having a unique SRB measure mu_t, and study the dependency o  mu_t on t. We shall start with the toy model of smooth expanding maps, for which mu_t has a smooth density which depends smoothly on t. We shall see that explicit formulas can be given for the derivatives of mu_t in this case. Then we shall recall (without proofs) results of Katok-Knieper-Pollicott-Weiss and Ruelle for  smooth uniformly hyperbolic dynamics, in particular Ruelle´s formula for the derivative of mu_t (as a distribution). The heart of the course will be devoted to the study of piecewise expanding unimodal maps, where we recently discovered with Daniel Smania that the SRB measure is differentiable if and only if f_t is tangent to the topolological class of f_0. In this case, we also obtain a formula for the derivative of the SRB measure.

Introduction to the Cohomological Theory of Dynamical Systems

Abstract in PDF format

Lorenz-like attractors.

In these lettures we shall see:
1. the equations of Lorenz and its main properties.
2. the geometric model for the equations of Lorenz
3. singular-hyperbolic attractors
4. a brief discussion about the proof that the flow generated by the above equations support a chaotic attractor with zero volume.
5. establish the equivalence between the geometrical model, the  flow generated by the equations of Lorenz and singular-hyperbolic  attractors.
6.  establish some topological properties displayed by  singular-hyperbolic attractors.

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Conservative dynamics and the calculus of variations.

Starting with elementary calculus of variations and Legendre transform, it is shown how the mathematical structures of conservative dynamics (Poincaré-Cartan integral  invariant, symplectic structure, hamiltonian form of the equations) arise from the simple computation of the variations of an action integral. The study of simple examples of integrable geodesic flows on the 2-torus then leads to the notion of Lagrangian submanifolds and to the Hamilton-Jacobi equation, whose relation to the Hamiltonian vector field is the first step of the duality between particles and waves.

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Flavio Abdenur
Ergodic Theory of C^1-Generic Non-Conservative Systems

We survey and discuss some recent progress (and also some ongoing work) on the ergodic theory of C^1-generic non-conservative systems. Topics include the existence of interesting invariant measures, the existence of physical attractors, and the relation between topological entropy and horseshoes.

Lorenzo Díaz

Non-generic measures for generic diffeomorphism

we discuss some recent resutls about homoclinic classes of $C^-1$-generic
diffeomorphis prove that non-hyperbolic homoclinic classes of these diffeomorphism
support an ergodic measure with uncountable support and zero Lyapunov

Neil Dobbs

 Pesin-Ledrappier theory for cusp maps

Pesin theory in the non-invertible setting was developed by Ledrappier for $C^{1+varepsilon}$ multimodal maps of the interval with non-flat critical points. This work can be extended to maps with flat critical points and with singularities where the derivative tends to infinity. As a corollary, the presence of very flat critical points precludes the existence of finite absolutely continuous probability measures.

Tomasz Downarowicz
Entropy properties of smooth maps
(with introduction to the theory of symbolic extensions)

Abstract in pdf

Vanderlei Horita
Destroying horseshoes via heterodimensional cycles

We propose a model for the destruction of three-dimensional horseshoes emph{via} heterodimensional cycles. This model yields some new dynamical features. Among other things, it provides examples of homoclinic classes properly contained in other classes and it is a model of a new sort of heteroclinic bifurcations we call emph{generating}.

Joint work with L. J. D´i az, I. Rios, and M. Sambarino

Roberto Markarian

Slow decay of correlations in billiard systems

We will present
a) some recent results on polynomial decay of
correlations in billiard systems and
b) a general strategy for estimating correlation functions
for smooth systems with singulariaties and weak hyperbolicity.

Alejandra Rodrigues Hertz
A new criterion for ergodicity and non-uniform hyperbolicity

We find a new criterion to establish ergodicity through the description of the ergodic components of an invariant measure. This criterion applies, for instance, to establish the C^1 Pugh-Shub conjecture about stable ergodicity for 2-dimensional center bundle. It is also used to prove uniqueness of SRB measures for transitive surface diffeomorphisms. The talk is aimed at a broad audience, and should be accessible to anyone with elementary notions in Pesin theory.

Federico Rodriguez Hertz
Non-uniform measure rigidity

Abstract: In this talk we shall present some recent results in
non-uniform measure rigidity for higher rank abelian actions. Measure
rigidity deals with the scarsity of invariant measures. For example,
Furstenberg raised the problem of studiyng the invariant measures for
the commuting endomorphisms of the circle, $ imes 2$ and $ imes 3$,
the unique invariant ergodic measures should be Lebesgue and the ones
supported over periodic orbits. This conjecture was solved in the case
the measure is assumed to be of possitive entropy with respect to an
element of the action in 1988 by R. Lyons and in 1990 by D. Rudolph.

This conjecture generalizes to a broad context. We will discuss this
generalizations and also some of the ingredients of its proof in the
simplest cases.

This is part of joint works with Boris Kalinin and Anatole Katok.

Raul Ures
Recent advances in partially Hiperbolic dynamics

Abstract in pdf