Related papers: Topological obstructions to dominated splitting fo…
We derive large deviations type (LDT) estimates for linear cocycles over an ergodic multifrequency torus translation. These models are called quasi-periodic cocycles. We make the following assumptions on the model: the translation vector…
We show that on a dense open set of analytic one-frequency complex valued cocycles in arbitrary dimension Oseledets filtration is either dominated or trivial. The underlying mechanism is different from that of the Bochi-Viana Theorem for…
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…
We present an analysis of one-dimensional models of dynamical systems that possess 'coherent structures'; global structures that disperse more slowly than local trajectory separation. We study cocycles generated by expanding interval maps…
The construction of spectral cocycle from the case of 1-dimensional substitution flows by Bufetov-Solomyak [arXiv:1802.04783] is extended to the setting of pseudo-self-similar tilings in ${\mathbb R}^d$, allowing expanding similarities with…
For substitution systems and translation flows, a new cocycle, which we call {\em spectral cocycle}, is introduced, whose Lyapunov exponents govern the local dimension of the spectral measure for higher-level cylindrical functions. The…
Given a linear cocycle over an ergodic homeomorphism on a compact metric space, we show that the existence of a uniform gap between the $p$-th and $(p+1)$-th Lyapunov exponent on a $C^0$-neighbourhood implies the existence of a dominated…
We consider extensions of Anosov diffeomorphisms of an infranilmanifold by the real vector space R^{\omega}. Our main result, based on the analogous theorem in finite dimensions proven by Nitica and Pollicott, is that any Holder cocycle…
We consider an m-dimensional analytic cocycle with underlying dynamics given by an irrational translation on the circle. Assuming that the d-dimensional upper left corner of the cocycle is typically large enough, we prove that the d largest…
One knows that the set of quasi-periodic Schr\"odinger cocycles with positive Lyapunov exponent is open and dense in the analytic topology. In this paper, we construct cocycles with positive Lyapunov exponent which can be approximated by…
We refine the theory of the cohomological equation for translation flows on higher genus surfaces with the goal of proving optimal results on the Sobolev regularity of solutions and of distributional obstructions. For typical translation…
We study cocycles taking values in the mapping class group of closed surfaces and investigate their leading topological Lyapunov exponent. Under a natural closing property, we show that the top topological Lyapunov exponent can be…
The paper is devoted to the properties of a complex matrix ``twisted,'' otherwise called ``spectral,'' cocycle, associated with substitution dynamical systems. Following a recent finding of Rajabzadeh and Safaee [arXiv:2501.16824] of an…
We develop a "local theory" of multidimensional quasiperiodic $\SL(2,\R)$ cocycles which are not homotopic to a constant. It describes a $C^1$-open neighborhood of cocycles of rotations and applies irrespective of arithmetic conditions on…
We show that a $\mathrm{GL}(d,\mathbb{R})$ cocycle over a hyperbolic system with constant periodic data has a dominated splitting whenever the periodic data indicates it should. This implies global periodic data rigidity of generic Anosov…
We demonstrate the existence of an open dense subset within the class of real analytic one-frequency quasi-periodic $\mathrm{\Sp}(4,\mathbb{R})$-cocycles, characterized by either the distinctness of all their Lyapunov exponents or the…
In this note, we show that if all Lyapunov exponents of a matrix cocycle vanish, then it can be perturbed to become cohomologous to a cocycle taking values in the orthogonal group. This extends a result of Avila, Bochi and Damanik to…
An asymptotic expansion is established for time averages of translation flows on flat surfaces. This result, which extends earlier work of A.Zorich and G.Forni, yields limit theorems for translation flows. The argument, close in spirit to…
In this m\'emoire we study quasiperiodic cocycles in semi-simple compact Lie groups. For the greatest part of our study, we will focus ourselves to one-frequency cocyles. We will prove that $C^{\infty}$ reducible cocycles are dense in the…
We consider cocycles of isometries on spaces of nonpositive curvature $H$. We show that the supremum of the drift over all invariant ergodic probability measures equals the infimum of the displacements of continuous sections under the…