Vanishing Exponents and Regularity of Aubry-Mather sets

Marie-Claude ARNAUD explained to me some of her results on the regularity of Aubry-Mather sets.

Alpern genericity theorem

Frédéric Le Roux has written a very lucid exposition of the Alpern genericity theorem. This theorem states that the same ergodic properties are generic in the set of volume-preserving measurable transformations and in the set of volume-preserving  homeomorphisms.

More precisely,  endow    with the weak topology, i.e., the coarsest generated by , for all measurable subsets and equip   with the uniform distance.

The theorem considers any   invariant under volume-preserving measurable isomorphisms. It states that   is a dense subset of if and only if is itself a dense subset of .

Positive Curvature for discrete spaces

Yann OLLIVIER gave a talk on his interpretation of Ricci curvature which sheds much light on this classical notion and allows its generalization, e.g., to discrete spaces.

He is interested in the rôle played by positive Ricci curvature in the concentration of the measure phenomenon discovered by Gromov in his generalization of Lévy’s theorem on 1-Lipschitz real functions on the unit N-sphere: if is a 1-Lipschitz function then, for all :

where is the natural measure on the sphere and is called the “observable diameter”.

He presented several striking applications (with coworkers) to some Markov chains, like the classical dynamics of the Ising model, as well as to the spectral radius of the Laplacian of a compact Riemannian manifold.

For a Riemannian manifold, the Ricci curvature along is defined by:

where , is the parallel transport of .

Thus positive Ricci curvature implies that “small balls are closer than their centers”.

This point of view generalizes to arbitrary Polish spaces endowed with local measures , . For any pair of points , the Ricci curvature is

where in the left hand side is Wasserstein L^1 distance:

with ranging over the couplings of and ranging over the 1-Lipschitz functions.

A positive Ricci curvature space is a space such that .

Example. with the geodesic distance from the underlying graph has postive Ricci curvature .

This applies to Markov chains with positive curvature, including the Ising model at sufficiently high temperature (higher than the critical temperature, maybe strictly higher). Some remarks of Dobrushin from the seventies may be rephrased in this geometric language.

It also allows estimating the spectral gap for some compact Riemannian manifolds. In particular, it gives a strengthening of Lichnérowiciz theorem in the case of variable curvature.

Orders of accumulation of entropy structures

Kevin McGoff gave a talk at Orsay on his work on the theory of entropy srtuctures and symbolic extensions. This theory was founded by Mike Boyle and Tomasz Downarowicz, among others.

This theory relates the continuity properties of the measure-theoretic entropy function with the existence of symbolic topological extensions with measure-theoretic entropies as close as possible to those of the initial system. This theory ascribes to any topological dynamical system an order of accumulation. M. Boyle and T. Downarowicz have shown that this is a countable ordinal.

David Burguet and K. McGoff have shown that any countable ordinal can be achieved by some topological dynamics. The proof relies on a realization theorem of T. Downarowicz and S. Serafin.

K. McGoff explained how he was able, by a more precise and direct construction to achieve the same on any prescribed compact manifold. The transformation can be chosen to be homeomorphic if the dimension is 2 or more.

Dilation factors of pseudo-Anosov homeomorphisms

Erwan Lanneau gave a talk in Orsay about his work on surfaces of translation. Pseudo-Anosov homeomorphisms are homeomorphisms of such surfaces which are affine away from the singular set of the surface and whose differential is hyperbolic. The dilation factor is the dominant eigenvalue of that differential. It is a Perron number.

The minimum of the dilation factor for given genus is known for geni 1 and 2 only (the techniques behind could be extended to genus 3 but not farther). One also knows that .

The main result of the talk is that the above does not hold when restricted to a given type of translation surfaces. More precisely, the moduli space of translation surfaces of given genus splits into connected components (at most three), one of them corresponding to hyperellipticity and the following holds:

Theorem (Boissy-Lanneau) Let be a pseudo-Anosov on a hyperelliptic translation surface of genus g admitting an involution with fixed points). Assume that has a unique singularity. Then its dilation is strictly greater than (but approach this value as ).

Rencontre DynNonHyp à Lille

Isabelle LIOUSSE organise la deuxième rencontre du projet DynNonHyp les 18 et 19 janvier prochains (lundi et mardi). Pour plus d’informations cliquez ici.

Snow in Orsay

A few pictures of the campus today

C^2 surface diffeomorphisms always have a symbolic extension

Most of topological dynamics studies systems of the form where is a continuous self-map and is a compact metric space. One approach is to “reduce” such systems to symbolic dynamical system, i.e., where is a closed subset of and such that .

J. Auslander asked about the obstructions for a topological system to have a symbolic extension, i.e., a symbolic system and a continuous surjection commuting with the dynamics: . There is an obvious one: a symbolic system (and therefore its topological factors) has finite topological entropy. Is there any other?

M. Boyle showed that this was indeed the case. With D. Fiebig and U. Fiebig, he showed that asymptotically h-expansive systems (including self-maps of compact manifolds by a result of mine based on Yomdin’s theory) always have a “nice” symbolic extension. T. Downarowicz and S. Newhouse showed that generic map have no symbolic extension whatever, leaving open the question of diffeomorphisms with finite smoothness.

T. Downarowicz and A. Maas showed that interval maps also always have symbolic extensions for .

David BURGUET has finally proved the same for arbitrary surface diffeomorphisms, see his preprint here.

Behind these works there is a rich and beautiful topological/ergodic/functional-analytical theory of entropy (called the entropy structure by T. Downarowicz) which does yet have the audience it deserves, in my opinion.

Discontinuity of the topological entropy for Lozi maps

I have shown that, like diffeomorphisms, piecewise affine surface homeomorphisms are approximated in entropy by horseshoes, away from their singularities. It follows in particular that their topological entropy is lower-semicontinuous: a small perturbation cannot cause a macroscopic drop in entropy.

The continuity of the entropy for such maps had been an open problem for some time. Rigorous numerical estimates by Duncan SANDS and Yutaka ISHII seemed to suggest some discontinuous drops, but investigation at a small scale suggested these drops to be steep yet continuous variations.

Izzet B. YILDIZ has solved this question by finding for Lozi maps on , numbers such that, for all with small enough:

• ;
• .

The verification turns out to be quite simple (that is, once you know where to look). The non-wandering set of is shown to be reduced to be reduced to the fixed points of its fourth iterates, yielding the zero entropy immediately. on the other hand is shown to admit 2 disjoint closed quadrilaterals such that hyperbolically crosses both and and hyperbolically crosses . This means that the sides of and can be branded alternatively s and u with the following property. The image of a u side crosses each of it meets, intersecting both their s sides and none of their u sides. This again yields the entropy estimate.

Uniformity for the decay of correlations

Decay of correlations at a speed given by some numbers for some dynamics and two spaces of functions over with zero average, is formulated in two similar ways in various works: for all ,

1. for some depending arbitrarily on ;
2. for some independent of ;

where is the correlation at time , a bilinear form .

The second type seems stronger than the first (and it is strictly so for arbitrary families of bilinear forms). However the above two types are really equivalent under very general assumptions, i.e., if (i) are Banach spaces and (ii) each is continuous for any given .

Indeed, fix .Let be defined by . Each such map is linear and continuous by (ii). Now, for each , . Using (i), the Banach-Steinhaus theorem gives such that . In other words, , defined by is linear and continuous.

Similarly defined by is linear and continuous. Now, for each , where is the unit ball in . Using (i), the Banach-Steinhaus theorem gives such that for all . But this implies the first type of decay above. CQFD.

This was pointed out to me by Sébastien GOUËZEL.

Added:

Jean-René CHAZOTTES points out that this is essentially Theorem B.1 of his paper joint with P. Collet and B. Schmitt: Statistical consequences of the Devroye inequality for processes. Applications to a class of non-uniformly hyperbolic dynamical systems. Nonlinearity 18 (2005).