Lower bounds for Hausdorff dimension

A classical theorem (Marstrand 1954) asserts that, given any Borel subset , the obvious inequality of the Hausdorff dimensions: is in fact an equality for almost all orthogonal projections . As is often the case it is usually very dificult to prove equality for a given projection.

Preliminary description: Michael HOCHMAN and Pablo SCHMERKIN have posted a preprint presenting a new method for doing this. They actually deal with the stronger statement involving the dimension of measures. The key step is a lower semicontinuity property under an assumption of regularity (the process defined by zooming at typical points must be stationary). The classical a.e. result then yields an open and dense set, which can be controlled using invariance of the considered measures under large groups.

Ergodic theory of sumsets

M. Bjorklund gave a talk in Orsay on the following problem in additive combinatorics:

Given , show that is “large” unless they have a “special structure”.

The size of sets is defined here in terms of their upper Banach density: where ranges over the integer sequences such that .

The special structure is the following form of quasi-periodicity: is a Bohr set if there exists a morphism mapping into some compact metric Abelian group and an open subset whose boundary has zero Haar measure such that .

His results (joint with A. Fish) are the following:

Theorem 1. If is a Bohr set then .

Theorem 2. If is a Bohr set and then is also a Bohr set.

The results are deduced from the following ergodic theorems:

Theorem 3. In the above setting, where ranges over the measurable subsets Haar measure $\lambda(F)\geq d^*(S)$.

Theorem 4. Let be an ergodic probabilistic dynamical system. Let and be as above. Let be such that . Then:

• belongs to the Kronecker factor of (it is measurable wrt the -algebra generated by the eigenfunctions of );
• is the unit circle and is an interval;
• there is a factor map , with a rotation, and the preimage of an interval up to a negligible set.

More precisely Theorems 1 and 2 are deduced from their ergodic counterparts using a slight strenthening of Furstenberg’s Correspondence principle and results on subsets of metric compact Abelian groups, notably due to Kneser in the fifties).

Stable ergodicity and partial hyperbolicity

Ya. Pesin gave a talk (Chevaleret, June 19, 2009) summarizing the work on Push-Shub Stable Ergodicity Conjecture and presenting some new results of his in the non-conservative case.

This conjecture states that there is a -dense and open set of ergodic diffeomorphisms among those which are partially hyperbolic and volume preserving. Note the tension between perturbative techniques which are mainly known in the setting and smooth ergodic theory in the setting.

One tries to establish accessibility, i.e., that it is possible to link any two points following alternatively stable and unstable manifolds. This is now known to hold:

• -densely according to Dolgopyat and Wilkinson;
• if the central dimension is 1, -densely according to Didier () and Rodriguez-Hertz, Roderiguez-Hertz and Ures.

Ergodicity then follows under smoothness and center-bunching (i.e., near conformality of the central bundle) according to Burns and Wilkinson and Rodriguez-Hertz, Roderiguez-Hertz and Ures.

The talk then turned to the non-conservative case, first recalling the result of Burns, Dolgopyat and Pesin establishing stable ergodicity provided: (i) all the exponents are strictly negative on a set of positive volume; (ii) smoothness; (iii) essential accessibility.

More precisely, is a partially hyperbolic, diffeomorphism of a compact manifold with a topological attractor . The role of the volume (no longer preserved) is played by u-measures.

They have been built “from the outside” by pushing the volume by Pesin-Sinai and Bonatti-Diaz-Viana. They have been characterized “from the inside”, by pushing the leaf volume by Pesin-Sinai. According to Bonatti-Diaz-Viana, any physical measure is a u-measure. According to Dolgopyat, a unique u-measure is a physical measure.

It is not difficult to see that uniqueness of the u-measure may fail even for topologically transitive attractor.

Pesin stated three theorems:

Theorem A.

1. There exists , a u-measure as soon as the Lyapunov exponents along the central bundle are all strictly negative and all unstable leaves are dense in the attractor.
   
2. if is unique as a u-measure, it is unique as an S.R.B. measure.
3. is Bernoulli.

This is related to results of Alvès-Bonatti-Viana. It applies to some examples with central dimension 1 (otherwise the sum of the central exponents is negative and this is a problem).

Theorem B.

In the setting of Theorem A, any -diffeomorphism which is -close to :

1. automatically satisfies the negativity of its central exponents;
2. has a unique u-measure (which is SRB);
3. this has full basin and is Bernoulli.

It is interesting to note that this theorem fails if one instead assumes that the central exponents are strictly positive. The authors do not claim that all unstable leaves are dense.

Theorem C.

Let be a -diffeomorphism with a partially hyperbolic attractor . Assume (i) there is a unique u-measure with strictly positive central exponents -almost everywhere; (ii) all unstable leaves are dense in . Then is stably ergodic.

This uses the result by Dolgopyat that the limit of a sequence of u-measures for a sequence of partially hyperbolic diffeomorphisms is itself a u-measure for the limit diffeomorphism (when they both exist). It also uses Pliss Lemma to get mesoscopic local central disk.

It is noted that they are few results in the presence of critical points (see Viana’s multidimensional attractors, however).

It is also observed that integrability is known to be breakable by a -perturbation (recall the result of Dolgopyat-Wilkinson). This is unknown in the -topology, except for central dimension 1 where a result of JP Marco exists.

Invisible Attractors

Yu. Ilyashenko and A. Negut have discovered the following dynamical phenomenon. Recall that the statistical attractor is the smallest closed set such that almost every orbit spends almost all its time arbitrarily close to . They say that an open set is -invisible if almost every orbit spends a fraction of its time less than in it.

-invisibility for very small may occur for trivial reasons: if is has a small intersection with the attractor, if the transformation (or its inverse) has very large Lipschitz constant, if it is close to a structurally unstable dynamics.

Ilyashenko with Negut have built non-trivial examples of such invisible subsets. For every integer parameter , they obtain a ball of radius set among a class of skew-products over the doubling map of the circle such that:

• the Lipschitz constant is independent of ;
• the map is structurally stable;
• the attractor covers ;
• the whole space above is -invisible

Ilyashenko and Volk have just published new examples with -invisibility.

Non-convergence des mesures de Gibbs pour T->0

Soit un système dynamique hyperbolique topologiquement transitif muni d’un potentiel continu . La mesure de Gibbs à température est par définition l’unique mesure de probabilité -invariante maximisant l’énergie libre (appelée pression en dynamique…): où est l’entropie mesurée.

Il est facile de voir que tout point d’accumulation de pour maximise . Pour certains systèmes, il existe plusieurs mesures maximisantes. Toutefois, Julien BREMONT a montré en 2001 que les convergent si est un sous-décalage de type fini et est localement constant. Depuis, plusieurs chercheurs ont cherché à généraliser ceci au cadre habituel, à savoir hölderienne.

Dans un récent préprint, Jean-René Chazottes et Mike HOCHMAN ont construit un potentiel lipschitzien pour lequel cette convergence n’a pas lieu. Ils montrent de plus que pour les sous-décalages de type fini multi-dimensionnels, cette convergence peut même tomber en défaut pour des potentiels localement constants!

Engendrement d’un plan irrationnel par des règles locales

Xavier BRESSAUD a présenté à Orsay une caractérisation locale du plan de Tribonacci.

La substitution de Tribonacci a été introduite par Gérard RAUZY. Il s’agit de . La matrice correspondante a un espace contractant de dimension 2. La frontière supérieure de l’union des cubes entiers intersectant ce plan est le plan discret de Tribonacci. En projection sur le plan contractant, on obtient un pavage du plan par tuiles en forme de parallèlogrammes de trois types. Par translation et fermeture dans une certaine topologie on obtient un système dynamique appelé espace de pavage.

Théorème. Quitte à colorier les tuiles, les pavages précédents sont caractérisés par des conditions d’adjacence.

La preuve est une adaptation des travaux de MOZES et de GOODMAN-STRAUSS. Le plan discret forme un système substitutif: les tuiles se regroupent par groupes de trois, etc. Il s’agit d’encoder cette structure hiérarchique par 1°) des chenaux liant les groupes d’ordre n formant un groupe d’ordre n+1; 2°) un marquage périphérique qui garantit le bon recollement des groupes de même ordre.

Flat limit in the Feigenbaum functional equation

G. SWIATEK explained in Orsay his joint work with G. LEVIN on the functional equation introduced by Feigenbaum for the study of the period-doubling “path to chaos” in the quadratic family. For each finite , they consider . A result of McMULLEN says that the solution is unique for any even integer . ECKMANN-WITWER showed the (uniform) convergence towards some with a flat critical point using computer-assisted estimates.

G. SWIATEK explained their new proof of the following strengthening:

1. extends analytically to the union of two topological disks in the complex plane containing respectively and and both mapped to a disk . The leading singularity is . Elsewhere the map has negative Schwarzian derivative.
2. has a Julia set with the usual characteristic properties (normality, density of periodic points, boundedness of the infinite orbits) which has zero area and full Hausdorff dimension, i.e., 2.
3. In the limit , the area and dimension of the corresponding Julia sets converge to the above values.

One of the motivations of this work is the well-known problem of Julia sets with positive area.

Question: Can one prove that these Julia sets have zero area (perhaps for large enough)?

Remark: One expects that the dimensions for finite $\ell$ are strictly between $1$ and $2$.

The proof follows McMULLEN’s: compactness (here with a uniform domain of analyticity); existence of uniformly quasi-conformal conjugacies between solutions (here bounded geometry has to be replaced by the presentation functions of LEDRAPPIER and MISIUREWICZ); towers, ie, sequences ; rigidity of towers yielding linear conjugacies
.

What we (don’t) know about Lozi

Duncan SANDS gave a talk for the Journée Affine Par Morceaux on the dynamics of Lozi maps. These are the piecewise affine homeomorphisms of the plane of the form where . Lozi introduced them as a toy model for the Hénon map, observing numerically some kind of strange attractor for . SANDS and ISHII have especially studied their topological entropy. The following picture shows what is known and what is not in the parameter plane:

In the grey area the entropy is known to be zero. In the turquoise area it is known to be positive (and maximal, ie, log 2, in the hatched part). In the white area in-between, there are examples with positive entropy (on for the part below the axis) but otherwise little is known.

The existence of a physical measure has been established only for a small part of the Misiurewicz triangle (for which a strange attractor is known). There is a larger triangle in which a simple trapping region exists.

Compléments pour MAT331

voir la page Enseignement

Bratelli diagrams, Vershik maps and Invariant measures

DEFINITION. A Bratelli diagram is a directed graph with a distinguished vertex such that (i) any vertex can be joined from by at least one path; (ii) such paths have all the same length called the level of ; (iii) there is a finite, non-zero number of arrows leaving each vertex.

Remark. Property (ii) is equivalent to the fact that there are parititions of the set of vertices and the set of arrows such that each arrow in $E_i$ goes from to .

DEFINITION. An order on a Bratelli diagram is the data, for each vertex , of a total order on the set of all arrows pointing to . A path is maximal, resp. minimal, if each of its arrows is maximal, resp. minimal, among the set of arrows with the same target. will denote the set of maximal, minimal paths.

To each Bratelli diagram is attached the set of infinite path starting at . As a subset of , it is compact.

DEFINITION. A Vershik map (or adic transformation) for an ordered Bratelli diagram is a homeomorphism with the following properties: (i) ; (iii) where is the smallest integer such that is not maximal and is the arrow following in the diagram order and is the minimal path joining to the origin of .

Remark. Not all ordered Bratelli diagrams admit Vershik maps (K. Medynets). If the set $X_{max}$ has empty interior, then there is at most one Vershik map.

The recent preprint of S. Bezugly, J. Kwiatkowski, K. Medynets and B. Solomyak describe the invariant measures under a Borel version of the Vershik map