Matrix Cocycles Research Articles

Nonexistence of Lyapunov exponents for matrix cocycles

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 strong contrast, for any dynamical system $f:X\rightarrow X$ with exponential specification property and a H$\ddot{\text{o}}$lder continuous matrix cocycle $A:X\rightarrow G (m,\mathbb{R})$, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of $A$ is residual (i.e., containing a dense $G_\delta$ set).

The entropy of Lyapunov-optimizing measures of some matrix cocycles

We consider one-step cocycles of $2 \times 2$ matrices, and we are interested in their Lyapunov-optimizing measures, i.e., invariant probability measures that maximize or minimize a Lyapunov exponent. If the cocycle is dominated, that is, the two Lyapunov exponents are uniformly separated along all orbits, then Lyapunov-optimizing measures always exist and are characterized by their support. Under an additional hypothesis of nonoverlapping between the cones that characterize domination, we prove that the Lyapunov-optimizing measures have zero entropy. This conclusion certainly fails without the domination assumption, even for typical one-step $\mathrm{SL}(2,\mathbb{R})$-cocycles; indeed we show that in the latter case there are measures of positive entropy with zero Lyapunov exponent.

Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices

We discuss spreading estimates for dynamical systems given by the iteration of an extended CMV matrix. Using a connection due to Cantero–Grünbaum–Moral–Velázquez, this enables us to study spreading rates for quantum walks in one spatial dimension. We prove several general results which establish quantitative upper and lower bounds on the spreading of a quantum walk in terms of estimates on a pair of associated matrix cocycles. To demonstrate the power and utility of these methods, we apply them to several concrete cases of interest. In the case where the coins are distributed according to an element of the Fibonacci subshift, we are able to rather completely describe the dynamics in a particular asymptotic regime. As a pleasant consequence, this supplies the first concrete example of a quantum walk with anomalous transport, to the best of our knowledge. We also prove ballistic transport for a quantum walk whose coins are periodically distributed.

Stochastic Stability of Lyapunov Exponents and Oseledets Splittings for Semi‐invertible Matrix Cocycles

We establish (i) stability of Lyapunov exponents and (ii) convergence in probability of Oseledets spaces for semi‐invertible matrix cocycles subjected to small random perturbations. The first part extends results of Ledrappier and Young to the semi‐invertible setting. The second part relies on the study of evolution of subspaces in the Grassmannian; the analysis developed here is based on higher‐dimensional Möbius transformations and is likely to be of wider interest. © 2015 Wiley Periodicals, Inc.

Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles

The pressure function $P(A, s)$ plays a fundamental role in the calculation of the dimension of "typical" self-affine sets, where $A=(A_1,\ldots, A_k)$ is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on $A$. As a consequence, we show that the dimension of "typical" self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated by the norm of matrix products or generalized singular value functions for matrix cocycles, and obtain applications on the continuity of equilibrium measures and the Lyapunov spectrum of matrix cocycles.

METASTABILITY, LYAPUNOV EXPONENTS, ESCAPE RATES, AND TOPOLOGICAL ENTROPY IN RANDOM DYNAMICAL SYSTEMS

We explore the concept of metastability in random dynamical systems, focusing on connections between random Perron–Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The Lyapunov spectrum of the random Perron–Frobenius cocycle and the random adjacency matrix cocycle is used to decompose the random system into two disjoint random systems with rigorous upper and lower bounds on (i) the escape rate in the setting of random maps, and (ii) topological entropy in the setting of random shifts of finite type, respectively.

Almost reduction and perturbation of matrix cocycles

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 general base dynamics and arbitrary dimension. We actually prove a fibered version of this result, and apply it to study the existence of dominated splittings into conformal subbundles for general matrix cocycles.

The generalised Berger–Wang formula and the spectral radius of linear cocycles

Using ergodic theory we prove two formulae describing the relationships between different notions of joint spectral radius for sets of bounded linear operators acting on a Banach space. The first formula was previously obtained by V.S. Shulman and Yu.V. Turovskiĭ using operator-theoretic ideas. The second formula shows that the joint spectral radii corresponding to several standard measures of noncompactness share a common value when applied to a given precompact set of operators. This result may be seen as an extension of classical formulae for the essential spectral radius given by R. Nussbaum, A. Lebow and M. Schechter. Both results are obtained as a consequence of a more general theorem concerned with continuous operator cocycles defined over a compact dynamical system. As a byproduct of our method we answer a question of J.E. Cohen on the limiting behaviour of the spectral radius of a measurable matrix cocycle.

Livšic Theorem for matrix cocycles

We prove the Liv sic Theorem for arbitrary GL( m;R) cocycles. We con- sider a hyperbolic dynamical system f : X! X and a Holder continuous function A : X! GL(m;R). We show that if A has trivial periodic data, i.e. A(f n 1 p) A(fp)A(p) = Id for each periodic point p = f n p, then there exists a Holder continuous function C : X ! GL(m;R) satisfying A(x) = C(fx)C(x) 1 for all x2 X. The main new ingredients in the proof are results of independent interest on relations between the periodic data, Lyapunov exponents, and uniform estimates on growth of products along orbits for an arbitrary Holder function A.

Recurrence of matrix cocycles

This paper considers measurable cocycles with values in the subgroup of SL(2, ℂ) of diagonal and skew-diagonal matrices over an ergodic transformation preserving the probability measure. We prove the recurrence of such cocycles under certain conditions as well as the equivalence of two definitions of the recurrence.

A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory

We use ergodic theory to prove a quantitative version of a theorem of M.A. Berger and Y. Wang, which relates the joint spectral radius of a set of matrices to the spectral radii of finite products of those matrices. The proof rests on a structure theorem for continuous matrix cocycles over minimal homeomorphisms having the property that all forward products are uniformly bounded.

Transfer operator, topological entropy and maximal measure for cocyclic subshifts

Cocyclic subshifts arise as the supports of matrix cocycles over a full shift and generalize topological Markov chains and sofic systems. We compute the topological entropy of a cocyclic subshift as the logarithm of the spectral radius of an appropriate transfer operator and give a concrete description of the measure of maximal entropy in terms of the eigenvectors. Unlike in the Markov or sofic case, the operator is infinite-dimensional and the entropy may be a logarithm of a transcendental number.

Ergodic reduction of random products of two-by-two matrices

We consider a random product of two-by-two matrices of determinant one over an abstract dynamical system. When the two Lyapunov exponents are distinct, Oseledets’ theorem asserts that the matrix cocycle is cohomologous to a diagonal matrix cocycle. When they are equal, we show that the cocycle is conjugate to one of three cases: a rotation matrix cocycle, an upper triangular matrix cocycle, or a diagonal matrix cocycle modulo a rotation by π/2.

Normal Forms for Random Differential-Equations

Given a dynamical system (Ω, F , P , θ( t)) and a random differential equation ẋ = ƒ(θ t ω, x) in R d with ƒ(ω, 0) = 0 a.s. The normal form problem is to construct a smooth near identity nonlinear random coordinate transformation h(ω) to make g(θ t ω, y) := Dh(θ t ω, y) −1 (ƒ(θ t ω, h(θ t ω, y)) − ( d/ dt) h(θ t ω, y)) as simple as possible, preferably linear. The linearization Dƒ(θ t , ω, 0) =: A(θ t ω) generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eifenvalues (Lyapunov exponents) and eigenspnces. Now the development runs pretty much parallel to the deterministic one, the difference being that the appearance of θ t turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the concept of ϵ-normal form necessary. The stochastic versions of resonance and averaging are developed. One- and two-dimensional examples are treated in detail.

Evolutionary Formalism for Products of Positive Random Matrices

We present a formalism to investigate directionality principles in evolution theory for populations, the dynamics of which can be described by a positive matrix cocycle (product of random positive matrices). For the latter, we establish a random version of the Perron-Frobenius theory which extends all known results and enables us to characterize the equilibrium state of a corresponding abstract symbolic dynamical system by an extremal principle. We develop a thermodynamic formalism for random dynamical systems, and in this framework prove that the top Lyapunov exponent is an analytic function of the generator of the cocycle. On this basis a fluctuation theory for products of positive random matrices can be developed which leads to an inequality in dynamical entropy that can be interpreted as a directionality principle for the mutation and selection process in evolutionary dynamics.

Normal forms for random diffeomorphisms

Given a dynamical system (Ω,ℱ, ℙ,θ) and a random diffeomorphism ϕ(ω): ℝd → ℝd with fixed point at x=0. The normal form problem is to construct a smooth near-identity nonlinear random coordinate transformation h(ω) to make the random diffeomorphism $$\tilde \varphi $$ (ω)=h(θω)−1○ϕ(ω)○ h(ω) “as simple as possible,” preferably linear. The linearization Dϕ(ω, 0)=:A(ω) generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eigenvalues (Lyapunov exponents) and eigenspaces. Now the development runs pretty much parallel to the deterministic one, the difference being that the appearance ofθ turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the conceptofe-normal form necessary. The stochastic versions of resonance and averaging are developed. The case of simple Lyapunov spectrum is treated in detail.

Stability of Lyapunov exponents

AbstractWe consider small random perturbations of matrix cocycles over Lipschitz homeomorphisms of compact metric spaces. Lyapunov exponents are shown to be stable provided that our perturbations satisfy certain regularity conditions. These results are applicable to dynamical systems, particularly to volume-preserving diffeomorphisms.

Random perturbations of matrix cocycles

AbstractLyapunov exponents for two-dimensional matrix cocycles are shown to be stable under certain stochastic perturbations.