Related papers: A positive proportion Livshits theorem

We consider the dependence on parameters of the solutions of cohomology equations over Anosov diffeomorphisms. We show that the solutions depend on parameters as smoothly as the data. As a consequence we prove optimal regularity results for…

Given a compact and complete metric space $X$ with several continuous transformations $T_1, T_2, \ldots T_H: X \to X,$ we find sufficient conditions for the existence of a point $x\in X$ such that $(x,x,\ldots,x)\in X^H$ has dense orbit for…

We consider transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We establish various properties of such systems including strong pinching, C^{1+\beta} smoothness of the…

Given a $C^2$- Anosov diffemorphism $f: M \rightarrow M,$ we prove that the jacobian condition $Jf^n(p) = 1,$ for every point $p$ such that $f^n(p) = p,$ implies transitivity. As application in the celebrated theory of Sinai-Ruelle-Bowen,…

In this paper, we first prove the strong Birkhoff Ergodic Theorem for subharmonic functions with the irrational shift on the Torus. Then, it is applied to the analytic quasi-periodic Jacobi cocycles. We show that if the Lyapunov exponent of…

In this paper, we prove that for $\mathcal{C}^1$ generic volume-preserving Anosov diffeomorphisms of a compact Riemannian manifold, Liv\v{s}ic measurable rigidity theorem holds. We also prove that for $\mathcal{C}^1$ generic…

We prove a generalization of Livsic's Theorem on the vanishing of the cohomology of certain types of dynamical systems. As a consequence, we strengthen a result due to Zimmer concerning algebraic hulls of Anosov actions of semisimple Lie…

In this article, we study the dynamics of geodesic flows on Riemannian (not necessarily compact) manifolds with no conjugate points. We prove the Anosov Closing Lemma, the local product structure, and the transitivity of the geodesic flows…

In a written correspondence with A. Livingston, Erd\H{o}s conjectured that for any arithmetical function $f$, periodic with period $q$, taking values in $\{-1,1\}$ when $q \nmid n$ and $f(n)=0$ when $q \mid n$, the series…

Let X be a vector field on a compact connected manifold M. An important question in dynamical systems is to know when a function g:M -> R is a coboundary for the flow generated by X, i.e. when there exists a function f: M->R such that Xf=g.…

We show that any topologically transitive codimension-one Anosov flow on a closed manifold is topologically equivalent to a smooth Anosov flow that preserves a smooth volume. By a classical theorem due to Verjovsky, any higher dimensional…

In this paper, we study the structure of Birkhoff spectra for hyperbolic dynamical systems. Given a H\"older observable \(f\) on a basic set \(\Lambda\), we obtain the following results: First, we characterize when the Birkhoff spectrum of…

We prove Liv\v{s}ic-type regularity results of coboundary representations for non-autonomous dynamical systems. Our results have an abstract nature and apply to several important specific situations, such as (higher-dimensional) random or…

We prove a nonuniformly hyperbolic version Liv\v{s}ic theorem, with cocycles taking values in the group of invertible bounded linear operators on a Banach space. The result holds without the ergodicity assumption of the hyperbolic measure.…

We prove the Kawamata-Viehweg vanishing theorem for a large class of divisors on surfaces in positive characteristic. By using this vanishing theorem, Reider-type theorems and extension theorems of morphisms for normal surfaces are…

Moser's theorem (1965) states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners. In…

A vanishing theorem for a convex cocompact hyperbolic manifold is established, which relates the L2 cohomology to the Hausdorff dimension of the limit set. The borderline case is shown to characterize the manifold completely.

In this work we obtain a Liouville theorem for positive, bounded solutions of the equation $$ (-\Delta)^s u= h(x_N)f(u) \quad \hbox{in }\mathbb{R}^{N} $$ where $(-\Delta)^s$ stands for the fractional Laplacian with $s\in (0,1)$, and the…

We construct non-Lipshitz flow in $H^s$ for the cubic nonlinear Schr\"odinger equation on the 2-torus of revolution with a Lipshitz or smooth metric . The non-Lipshitz property holds for all $s<2/3$ for Lipshitz metric and $s<1/2$ for…

We establish a relative version of Gromov's Vanishing Theorem in the presence of amenable open covers with small multiplicity, extending a result of Li, L\"oh, and Moraschini. Our approach relies on Gromov's theory of multicomplexes.