Related papers: Projective cocycles over SL(2,R) actions: measures…

We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2,R) is supported on an…

We study the Lyapunov spectrum of the Kontsevich--Zorich cocycle on $SL(2,\mathbb{R})$-invariant subbundles of the Hodge bundle over the support of a $SL(2,\mathbb{R})$-invariant probability measure on the moduli space of Abelian…

In this paper we study topological aspects of the dynamics of the foliated horocycle flow on flat projective bundles over hyperbolic surfaces and we derive ergodic consequences. If $\rho : \Gamma \to {\rm PSL}(n+1,\mathbb{R})$ is a…

Let $G$ be a semisimple Lie group acting on a space $X$, let $\mu$ be a compactly supported measure on $G$, and let $A$ be a strongly irreducible linear cocycle over the action of $G$. We then have a random walk on $X$, and let $T$ be the…

We prove that invariant subbundles of the Kontsevich-Zorich cocycle respect the Hodge structure. In particular, we establish a version of Deligne semisimplicity in this context. This implies that invariant subbundles must vary polynomially…

We prove that if a H\"older continuous fiber-bunched cocycle $\hat{A}$ over an invertible hyperbolic transitive shift $\hat{\Sigma}$ satisfies an appropriate strong irreducibility condition on Grassmannians, then $\hat{\Sigma}$ admits an…

We prove a geometric criterion on a $\SL$-invariant ergodic probability measure on the moduli space of holomorphic abelian differentials on Riemann surfaces for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle on the real…

We consider an ergodic invariant measure $\mu$ for a smooth action of $Z^k$, $k \ge 2$, on a $(k+1)$-dimensional manifold or for a locally free smooth action of $R^k$, $k \ge 2$ on a $(2k+1)$-dimensional manifold. We prove that if $\mu$ is…

We prove that if a Borel probability measure (\mu) on (\T) is invariant under the action of a "large" multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then (\mu) is either…

We prove the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for a measure supported on abelian differentials which come from non-orientable quadratic differentials through a standard orientating, double cover construction. The…

The celebrated Oseledets theorem \cite{O}, building over seminal works of Furstenberg and Kesten on random products of matrices and random variables taking values on non-compact semisimple Lie groups \cite{FK,Furstenberg}, ensures that the…

Let $G$ be a real semisimple Lie group with finite centre and without compact factors, $Q<G$ a parabolic subgroup and $X$ a homogeneous space of $G$ admitting an equivariant projection on the flag variety $G/Q$ with fibres given by copies…

We prove absolute continuity of "high entropy" hyperbolic invariant measures for smooth actions of higher rank abelian groups assuming that there are no proportional Lyapunov exponents. For actions on tori and infranilmanifolds existence of…

We prove that there is a residual subset $\mathcal{S}$ in $\text{Diff}^1(M)$ such that, for every $f\in \mathcal{S}$, any homoclinic class of $f$ with invariant one dimensional central bundle containing saddles of different indices (i.e.…

In the moduli space $H_g$ of normalized translation surfaces of genus $g$, consider, for a small parameter $\rho >0$, those translation surfaces which have two non-parallel saddle-connections of length $\leq \rho$. We prove that this subset…

It follows from Oseledec Multiplicative Ergodic Theorem that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In…

In the first part of this paper, we formulate a general setting in which to study the ergodic theory of differentiable $\mathbb{Z}^d$-actions preserving a Borel probability measure. This framework includes actions by $C^{1+\text{H\"older}}$…

This work aims to investigate the well-posedness and the existence of ergodic invariant measures for a class of third grade fluid equations in bounded domain $D\subset\mathbb{R}^d,d=2,3,$ in the presence of a multiplicative noise. First, we…

We prove several general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous action of the Lie group SL(2, R) on a locally compact space. These results are motivated by theorems of…

An analytic quasi-periodic cocycle is a linear cocycle over a fixed ergodic torus translation of one or several variables, where the fiber action depends analytically on the base point. Consider the space of all such cocycles of any given…