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Entropy of eigenfunctions of the Laplacian in dimension 2

Gabriel Rivière (2010)

Journées Équations aux dérivées partielles

We study asymptotic properties of eigenfunctions of the Laplacian on compact Riemannian surfaces of Anosov type (for instance negatively curved surfaces). More precisely, we give an answer to a question of Anantharaman and Nonnenmacher [4] by proving that the Kolmogorov-Sinai entropy of a semiclassical measure μ for the geodesic flow g t is bounded from below by half of the Ruelle upper bound. (This text has been written for the proceedings of the 37 èmes Journées EDP (Port d’Albret-June, 7-11 2010))

Ergodic theory of interval exchange maps.

Marcelo Viana (2006)

Revista Matemática Complutense

A unified introduction to the dynamics of interval exchange maps and related topics, such as the geometry of translation surfaces, renormalization operators, and Teichmüller flows, starting from the basic definitions and culminating with the proof that almost every interval exchange map is uniquely ergodic. Great emphasis is put on examples and geometric interpretations of the main ideas.

Explicit Teichmüller curves with complementary series

Carlos Matheus, Gabriela Weitze-Schmithüsen (2013)

Bulletin de la Société Mathématique de France

We construct an explicit family of arithmetic Teichmüller curves 𝒞 2 k , k ∈ ℕ , supporting SL ( 2 , ℝ ) -invariant probabilities μ 2 k such that the associated SL ( 2 , ℝ ) -representation on  L 2 ( 𝒞 2 k , μ 2 k ) has complementary series for every k ≥ 3 . Actually, the size of the spectral gap along this family goes to zero. In particular, the Teichmüller geodesic flow restricted to these explicit arithmetic Teichmüller curves 𝒞 2 k has arbitrarily slow rate of exponential mixing.

Exponential mixing for the Teichmüller flow

Artur Avila, Sébastien Gouëzel, Jean-Christophe Yoccoz (2006)

Publications Mathématiques de l'IHÉS

We study the dynamics of the Teichmüller flow in the moduli space of abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Hölder observables. A geometric consequence is that the S L ( 2 , ℝ ) action in the moduli space has a spectral gap.

Feuilletages transversalement projectifs sur les variétés de Seifert

Thierry Barbot (2003)

Annales de l’institut Fourier

Soit M une variété de Seifert de groupe fondamental non virtuellement résoluble. Soit Φ un feuilletage de dimension 1 sur M , muni d’une structure projective réelle transverse. On suppose que Φ satisfait la propriété de relèvement des chemins, i.e., que l’espace des feuilles du relèvement de Φ dans le revêtement universel de M est séparé au sens de Hausdorff. On montre qu’à revêtements finis près, Φ est soit une fibration projective, soit un feuilletage géodésique convexe, soit un feuilletage horocyclique...

Fourier expansion along geodesics on Riemann surfaces

Anton Deitmar (2014)

Open Mathematics

For an eigenfunction of the Laplacian on a hyperbolic Riemann surface, the coefficients of the Fourier expansion are described as intertwining functionals. All intertwiners are classified. A refined growth estimate for the coefficients is given and a summation formula is proved.

Generic measures for geodesic flows on nonpositively curved manifolds

Yves Coudène, Barbara Schapira (2014)

Journal de l’École polytechnique — Mathématiques

We study the generic invariant probability measures for the geodesic flow on connected complete nonpositively curved manifolds. Under a mild technical assumption, we prove that ergodicity is a generic property in the set of probability measures defined on the unit tangent bundle of the manifold and supported by trajectories not bounding a flat strip. This is done by showing that Dirac measures on periodic orbits are dense in that set.In the case of a compact surface, we get the following sharp result:...

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