Related papers: Density of positive Lyapunov exponents for symplec…
For any polynomial map with a single critical point, we prove that its lower Lyapunov exponent at the critical value is negative if and only if the map has an attracting cycle. Similar statement holds for the exponential maps and some other…
We consider some general classes of random dynamical systems and show that a priori very weak nonuniform hyperbolicity conditions actually imply uniform hyperbolicity.
We prove that Lyapunov exponents of typical H\"older continuous fiber bunched linear cocycles over singular hyperbolic flows have multiplicity 1: the subspace of exceptional cocycles has infinite codimension.
This paper is devoted to study stability of Lyapunov exponents and simplicity of Lyapunov spectrum for bounded random compact operators on a separable infinite-dimensional Hilbert space from a generic point of view generated by the…
In this paper we prove the continuity of all Lyapunov exponents, as well as the continuity of the Oseledets decomposition, for a class of irreducible cocycles over strongly mixing Markov shifts. Moreover, gaps in the Lyapunov spectrum lead…
In this paper we study ergodic optimization problems for typical cocycles. We consider one-step $SL(2,\mathbb{R})$-cocycles that satisfy pinching and twisting conditions. We prove that the Lyapunov maximizing measures have zero entropy…
We study the fiber Lyapunov exponents of step skew-product maps over a complete shift of $N$, $N\ge2$, symbols and with $C^1$ diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic,…
For linear cocycles over both maps and flows, we obtain Datko-Pazy type of conditions under which all Lyapunov exponents of a given cocycle are negative. Furthermore, by combining our results with the results on subadditive ergodic…
In 2019 Anthony Quas, Philippe Thieullen and Mohamed Zarrabi introduced the concept of strong fast invertibility for linear cocycles. It relates the growth of volumes between different initial times and, together with a condition on…
We present an example of a discontinuity point for the Lyapunov exponents when viewed as a function of the cocycle in a topology finer than the $C^0$-topology. The linear cocycle taking values in SL(2,R) is locally constant, defined over a…
It has been recently realized that for abundant dynamical systems on a compact manifold, the set of points for which Lyapunov exponents fail to exist, called the Lyapunov irregular set, has positive Lebesgue measure. In the present paper,…
We consider the linear cocycle $(T,A)$ induced by a measure preserving dynamical system $T:X \to X$ and a map $A:X \to \mathit{SL}(2,\mathbb{R})$. We address the dependence of the upper Lyapunov exponent of $(T,A)$ on the dynamics $T$ when…
We show that a linear Young differential equation generates a topological two-parameter flow, thus the notions of Lyapunov exponents and Lyapunov spectrum are well-defined. The spectrum can be computed using the discretized flow and is…
Consider the space of two dimensional random linear cocycles over a shift in finitely many symbols, with at least one singular and one invertible matrix. We provide an explicit formula for the unique stationary measure associated to such…
In this paper, we study the size of the level sets of all Lyapunov exponents. For typical cocycles, we establish a variational relation between the topological entropy of the level sets of Lyapunov exponents and the topological pressure of…
We study the Lyapunov exponents of a two-dimensional, random Lorentz gas at low density. The positive Lyapunov exponent may be obtained either by a direct analysis of the dynamics, or by the use of kinetic theory methods. To leading orders…
We consider the top Lyapunov exponent associated to a dissipative linear evolution equation posed on a separable Hilbert or Banach space. In many applications in partial differential equations, such equations are often posed on a scale 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…
We show that typical (in the sense of Bonatti-Viana) H\"{o}lder and fiber-bunched $GL_d(\mathbb{R})$-valued cocycles over a subshift of finite type are uniformly quasi-multiplicative with respect to all singular value potentials. We prove…
We consider discrete one-dimensional Schr\"odinger operators whose potentials are generated by sampling along the orbits of a general hyperbolic transformation. Specifically, we show that if the sampling function is a non-constant H\"older…