Construction of ergodic cocycles that are fundamental solutions to linear systems of a special form (original) (raw)

Change of velocity and ergodicity in flows and in Markov semi-groups

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1977

Let {T(t)}t>_ o be a strongly continuous semi-group of Markov operators on C(X) with generator G. If mE C(X) is strictly positive, mG generates a semigroup. If {T(t)} is a group given by a flow, m may have isolated zeros and, under some regularity conditions, mG will still generate a flow, constructed explicitly. The connection between some ergodic properties of the new and original flow is studied. For the Markov semi-groups, the new one is strongly ergodic if and only if the original one is strongly ergodic.

Recent Trends in Ergodic Theory and Dynamical Systems

Contemporary Mathematics, 2015

We discuss some of the issues that arise in attempts to classify automorphisms of compact abelian groups from a dynamical point of view. In the particular case of automorphisms of one-dimensional solenoids, a complete description is given and the problem of determining the range of certain invariants of topological conjugacy is discussed. Several new results and old and new open problems are described.

New conditions for (non)uniform behaviour of linear cocycles over flows

Journal of Mathematical Analysis and Applications, 2018

We give a characterization of tempered exponential dichotomies for linear cocycles over flows in terms of the spectral properties of certain linear operators. We consider noninvertible linear cocycles acting on infinite-dimensional spaces and our approach avoids the use of Lyapunov norms. Finally, we apply obtained results to give new condtions for uniform exponential stability of linear cocycles.

Ergodic properties of infinite extensions of area-preserving flows

Mathematische Annalen, 2011

We consider volume-preserving flows (Φ f t ) t∈R on S × R, where S is a closed connected surface of genus g ≥ 2 and (Φ f t ) t∈R has the form Φ f t (x, y) = φtx, y + t 0 f (φsx) ds where (φt) t∈R is a locally Hamiltonian flow of hyperbolic periodic type on S and f is a smooth real valued function on S. We investigate ergodic properties of these infinite measure-preserving flows and prove that if f belongs to a space of finite codimension in C 2+ǫ (S), then the following dynamical dichotomy holds: if there is a fixed point of (φt) t∈R on which f does not vanish, then (Φ f t ) t∈R is ergodic, otherwise, if f vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension (Φ 0 t ) t∈R . The proof of this result exploits the reduction of (Φ f t ) t∈R to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of (φt) t∈R on which f does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.

Reiterated Ergodic Algebras and Applications

Communications in Mathematical Physics, 2010

We redefine the homogenization algebras without requiring the separability assumption. We show that this enables one to treat more complicated homogenization problems than those solved by the previous theory. In particular we exhibit an example of algebra which, contrary to the algebra of almost periodic functions, induces no homogenization algebra. We prove some general compactness results which are then applied to the resolution of some homogenization problems related to the generalized Reynolds type equations and to some nonlinear hyperbolic equations.

Locally compact groups appearing as ranges of cocycles of ergodic -actions

Ergodic Theory and Dynamical Systems, 1985

The paper contains the proof of the fact that every solvable locally compact separable group is the range of a cocycle of an ergodic automorphism. The proof is based on the theory of representations of canonical anticommutation relations and the orbit theory of dynamical systems. The slight generalization of reasoning shows further that this result holds for amenable Lie groups as well and can be also extended to almost connected amenable locally compact separable groups.

Stable ergodicity and Anosov flows

Topology, 2000

In this note we prove that if M is a 3-manifold and ϕ t : M → M is a C 2 , volume-preserving Anosov flow, then the time-1 map ϕ 1 is stably ergodic if and only if ϕ t is not a suspension of an Anosov diffeomorphism.

Lyapunov non-typical behavior for linear cocycles through the lens of semigroup actions

2021

The celebrated Oseledets theorem \cite{O}, building over seminal works of Furstenberg and Kesten on random products of matrices and random variables taking values on non-compact semisimple Lie groups \cite{FK,Furstenberg}, ensures that the Lyapunov exponents of mathrmSL(d,mathbbR)\mathrm{SL}(d,\mathbb R)mathrmSL(d,mathbbR)-cocycles (dgeqslant2)(d\geqslant 2)(dgeqslant2) over the shift are well defined for all points in a total probability set, ie, a full measure subset for all invariant probabilities. Given a locally constant mathrmSL(d,mathbbR)\mathrm{SL}(d,\mathbb R)mathrmSL(d,mathbbR)-valued cocycle we are interested both in the set of points on the shift space for which some Lyapunov exponent is not well defined, and in the set of directions on the projective space mathbfPmathbbRd\mathbf P \mathbb R^dmathbfPmathbbRd along which there exists no well defined exponential growth rate of vectors for a certain product of matrices. We prove that if the semigroup generated by finitely many matrices in mathrmSL(d,mathbbR)\mathrm{SL}(d,\mathbb R)mathrmSL(d,mathbbR) is not compact and is strongly projectively accessible then there exists a dens...