A measure of maximal entropy (by A. Chéritat)

A measure of maximal entropy of a rational fraction on the Riemann sphere according to Arnaud Chéritat

Copula in Statistics

The cumulative distribution function of a random variable is: .

The copula of is such that: where , resp. , is the distribution function of , resp. . It is unique if each variable is continuous (atomless law).

Theorem (Sklar). A function is the copula of some random variable with values in if and only if the following properties are satisfied for all : (i) ; (ii) ; (iii) where for all.

Remark. A function is a copula iff it is the (restriction to of the) repartition function of a random vector where each is uniform over .

is independent iff it admits the copula .

For an uniform random variable , the vector admits the copula .

Fréchet bounds: Any copula satisfies:

The left hand side is itself a copula only for .

The restriction of a copula to fewer variables is again a copula.There is no known general way to extend a copula to more variables.

See wikipedia.

Oberwolfach

Finite codimension in Banach spaces

On 2009/11/27 at IHP on the occasion of the Katok 65 conference, Patrick BERNARD presented his notion of codimension in Banach spaces with applications to Mather measures and the transversality theorem.

B will denote the ambient Banach space. A large part of the theory extends to Fréchet spaces (things get tougher once C^1 smoothness is involved).

A Lipschitz graph in of codimension is a subset of the graph of a Lipschitz function where and . A rectifiable set of codimension is the image by the canonical projection of a Lipschitz graph of codimension in .  A subset of is of codimension if it is a countable union of rectifiable subsets of that codimension.

Theorem (Zajicek 2008). The complement of a subset of positive codimension in a Banach space is Baire generic (i.e., a dense ) and prevalent.

Codimension behaves as expected with respect to C^1 maps: it is invariant under C^1 diffeomorphisms, it decreases the Fredholm index of a C^1 map, it is preserved by preimage under submersions.

It is then strikingly straightforward to prove the following applications:

Theorem. The set of potentials with k+1 Mather measures has codimension k in C^p

Theorem. The set of maps from a m-dimensional manifold to that fail to be injective immersions has codimension n-2k

Abundance of weakly diophantine generators

Emmanuel BREUILLARD

Ergodicity of smooth systems with product measures

Federico RODRIGUEZ-HERTZ presented at the IHP conference for Katok’s 65th birthday several results pertaining to the ergodicity of systems with some product structure.

Theorem (RHRHTU 2008). Let be a C^2 diffeomorphism of a compact manifold which preserves volume. Let be a hyperbolic periodic point. Define has a transverse point . Define similarly.

If and then these two sets are equal (up to a zero volume set) and the restriction of there, is ergodic with no zero exponent.

Corollary. Ergodicity is C^1-open and dense within C^2 partially hyperbolic diffeomorphisms with central dimension at most 2.

Theorem (M.A. Rodriguez-Hertz). A generic 3-dimensional volume-preserving either has all its Lyapunov exponents vanishing Lebesgue-almost everywhere, or has a dominated splitting separating positive and negative Lyapunov exponents and is ergodic with respect to the volume.

Remark. The proof is expected to generalize to symplectomorphisms in arbitrary dimensions.

Remark. This is a stronger but less general result that that of AVILA and BOCHI according to which, in arbitrary dimension, the C^1-generic volume-preserving diffeomorphism either has a dominated splitting and a set of positive volume of non-uniformly hyperbolic points defining an ergodic component of the volume or there is, almost everywhere, at least one zero exponent.

Consider a bundle with fiber a smooth compact manifold and a bundle map acting smoothly on the fibers and preserving on the base a measure.

Theorem. Assume that there are well-defined holonomy maps on the fibers: for . If the exponents along the fiber vanish, then these holonomy preserve the conditional measures. If the base map is hyperbolic and the base measure has a product structure then the conditional measures depend continuously on the base point.

This extends a classical result of Ledrappier for matrix cocycles. The above has the following application:

Theorem (RHRHTU). Let   be partially hyperbolic with a central foliation into circles. If has the accessibility property then either is topologically conjugate to an isometric extension of an Anosov diffeomorphism (and has a unique maximal entropy measure which has a zero central exponent), or has at least 2 maximal entropy measures and finitely many of them and their Lyapunov exponents are bounded away from zero.

Question. Is the set of diffeomorphism with finitely many ergodic maximal entropy measures C^r-dense?

Actions of the multiplicative integers

Vitaly BERGELSON

Spectral dichotomy for one-frequency Schrodinger operators

Artur AVILA

Orbit closures of Zariski dense subgroups of SL(n,R)

Yves BENOIST and Jean-François QUINT

Ergodicity of partially hyperbolic symplectomorphisms

Amie WILKINSON presented new results towards the Pugh-Shub Stable ergodicity conjecture. In particular, with A. AVILA and J. BOCHI, she proved that ergodicity is generic in C^1 partially hyperbolic symplectomorphisms. She noted that, by a result of SAGHIN and Z. XIA, a stably ergodic symplectomorphism is automatically partially hyperbolic (which fails for conservative diffeomorphisms by an example of A. TAHZIBI).

The ingredients of the proof are:

• Bochi’s alternative: a symplectomorphism which is not Anosov has only zero exponents Lebesgue almost everywhere.
• A new criterion for ergodicity and saturation for C^2 partially hyperbolic diffeomorphisms  involving a new, non-uniform variant of the center bunching property (used to prove that a set is simultaneously saturated wrt to the stable and unstable foliations)
• A perturbation technique introduced by M.-C. ARNAUD in her proof of Mañé’s ergodic closing lemma
• The ergodic  diffeomorphisms of the disk arbitrarily close to the identity, built by ANOSOV and KATOK
• A Baire argument