Alex Eskin
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View article: Measure rigidity for generalized u-Gibbs states and stationary measures via the factorization method
Measure rigidity for generalized u-Gibbs states and stationary measures via the factorization method Open
We obtain measure rigidity results for stationary measures of random walks generated by diffeomorphisms, and for actions of $\operatorname{SL}(2,\mathbb{R})$ on smooth manifolds. Our main technical result, from which the rest of the theore…
View article: Continuity of the Lyapunov exponents of random matrix products
Continuity of the Lyapunov exponents of random matrix products Open
We prove that the Lyapunov exponents of random products in a (real or complex) matrix group depends continuously on the matrix coefficients and probability weights. More generally, the Lyapunov exponents of the random product defined by an…
View article: Geometric properties of partially hyperbolic measures and applications to measure rigidity
Geometric properties of partially hyperbolic measures and applications to measure rigidity Open
We give a geometric characterization of the quantitative non-integrability, introduced by Katz, of strong stable and unstable bundles of partially hyperbolic measures and sets in dimension 3. This is done via the use of higher order templa…
View article: Effective counting of simple closed geodesics on hyperbolic surfaces
Effective counting of simple closed geodesics on hyperbolic surfaces Open
We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most L on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponenti…
View article: Self-joinings for 3-IETs
Self-joinings for 3-IETs Open
We show that typical interval exchange transformations on three intervals are not 2-simple answering a question of Veech. Moreover, the set of self-joinings of almost every 3-IET is a Poulsen simplex.
View article: Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group
Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group Open
We consider the action of $SL(2,\\mathbb{R})$ on a vector bundle $\\mathbf{H}$\npreserving an ergodic probability measure $\\nu$ on the base $X$. Under an\nirreducibility assumption on this action, we prove that if $\\hat\\nu$ is any\nlift…
View article: Möbius disjointness for interval exchange transformations on three intervals
Möbius disjointness for interval exchange transformations on three intervals Open
We show that Sarnak's conjecture on Mobius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.
View article: Maryam Mirzakhani: 1977–2017
Maryam Mirzakhani: 1977–2017 Open
Curt McMullen was widely acclaimed and contained already the seeds of what would become her first three major papers.All three of these results-a new proof of Witten's conjecture, a computation of the volume of the moduli space of curves, …
View article: A. Eskin - Counting closed geodesics on translation surfaces
A. Eskin - Counting closed geodesics on translation surfaces Open
We show that the number of closed geodesics in the flat metric on a translation surface of length at most R is asymptotic to e hR / (hR). This is joint work with Kasra Rafi
View article: The algebraic hull of the Kontsevich–Zorich cocycle
The algebraic hull of the Kontsevich–Zorich cocycle Open
We compute the algebraic hull of the Kontsevich-Zorich cocycle over any GL^+_2(R) invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.
View article: Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group
Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group Open
We consider the action of $SL(2,\mathbb{R})$ on a vector bundle $\mathbf{H}$ preserving an ergodic probability measure $ν$ on the base $X$. Under an irreducibility assumption on this action, we prove that if $\hatν$ is any lift of $ν$ to a…
View article: Large-scale rank of Teichmüller space
Large-scale rank of Teichmüller space Open
Let X be quasi-isometric to either the mapping class group equipped with the\nword metric, or to Teichmuller space equipped with either the Teichmuller\nmetric or the Weil-Petersson metric. We introduce a unified approach to study\nthe coa…
View article: Mobius disjointness for interval exchange transformations on three\n intervals
Mobius disjointness for interval exchange transformations on three\n intervals Open
We show that Sarnak's conjecture on Mobius disjointness holds for interval\nexchange transformations on three intervals (3-IETs) that satisfy a mild\ndiophantine condition.\n