Related papers: Markov extensions and lifting measures for complex…
We consider orthogonally invariant probability measures on $\mathrm{GL}_n(\mathbb{R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices to the Lyapunov exponents of random matrix products independently drawn…
We consider skew-products of quadratic maps over certain Misiurewicz-Thurston maps and study their statistical properties. We prove that, when the coupling function is a polynomial of odd degree, such a system admits two positive Lyapunov…
Let $P(\Delta)$ be a polynomial of the Laplace operator $\Delta=\sum_{j=1}^n\frac{\partial^2}{\partial x^2_j}$ on $\mathbb{R}^n$. We prove the existence of weak solutions of the equation $P(\Delta)u=f$ and the existence of a bounded right…
We establish large deviation type estimates for i.i.d. products of two dimensional random matrices with finitely supported probability distribution. The estimates are stable under perturbations and require no irreducibility assumptions. In…
We consider a finite family of invertible $2 \times 2$ real matrices and a transitive Markov shift on the index set. Let $\lambda$ be the top Lyapunov exponent for random matrix products driven by the Markov shift. We prove that, if the…
We use height arguments to prove two results about the dynamical Mordell-Lang problem. (i) For an endomorphism of a projective variety, the return set of a dense orbit into a curve is finite if any cohomological Lyapunov multiplier of any…
We consider the action of $SL(2,\mathbb{R})$ on a vector bundle $\mathbf{H}$ preserving an ergodic probability measure $\nu$ on the base $X$. Under an irreducibility assumption on this action, we prove that if $\hat\nu$ is any lift of $\nu$…
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…
By using the integration by parts formula of a Markov operator, the closability of quadratic forms associated to the corresponding invariant probability measure is proved. The general result is applied to the study of semilinear SPDEs,…
We study quadratic skew-products with parameters driven over piecewise expanding and Markov interval maps with countable many inverse branches, a generalization of the class of maps introduced by Viana. In particular we construct a class of…
The expansion exponent (or expansion constant) for maps was introduced by Schreiber in \cite{s}. In this paper, we introduce the analogous exponent for measures. We shall prove the following results: The expansion exponent of a measurable…
We introduce the Markov extension, represented schematically as a tower, to the study of dynamical systems with holes. For tower maps with small holes, we prove the existence of conditionally invariant probability measures which are…
Bobkov (J. Theoret. Probab. 18(2) (2005) 399-412) investigated an approximate de Finetti representation for probability measures, on product measurable spaces, which are symmetric under permutations of coordinates. One of the main results…
We consider the product of i.i.d. random matrices sampled according to a probability measure $\mu$ supported on a strongly irreducible and proximal subset of a compact set $S\subset GL(d,\mathbb{R})$. We establish the local analyticity of…
A version of Markov's estimate for the derivative of a polynomial is proved with the interval [-1,1] replaced by an arbitrary continuum in the complex plane.
By using Lyapunov conditions, weak Poincar\'e inequalities are established for some probability measures on a manifold $(M,g)$. These results are further applied to the convolution of two probability measures on $\R^d$. Along with explicit…
We present conservativeness criteria for sub-Markovian semigroups generated by divergence type operators with specified infinitesimally invariant measures. The conservativeness criteria in this article are derived by $L^1$-uniqueness and…
We study the top Lyapunov exponent of a product of random $2 \times 2$ matrices appearing in the analysis of several statistical mechanical models with disorder, extending a previous treatment of the critical case (Giacomin and Greenblatt,…
We consider a family of Markov chains whose transition dynamics are affected by model parameters. Understanding the parametric dependence of (complex) performance measures of such Markov chains is often of significant interest. The…
On a one-sided shift of finite type we prove that for a generic Holder continuous function there is a unique maximizing measure. We show that b-Holder continuous functions can be approximated in the a-Holder topology, a<b, by a function…