Spectral, ergodic and cohomological prolems in dynamical systems

2016

Abstract. In this paper the canonical representation of an A-contraction T on a Hilbert space H is used to obtain some conditions concerning the concept of A-ergodicity studied in [14–17]. The regular case and the case of R(A) closed are considered, and specifically, the TT ∗-contractions are studied. Some spectral properties are also given for certain particular class of A-isometries.

2012

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We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.

Long Time Behaviour of Classical and Quantum Systems, 2001

Mathematics of Complexity and Dynamical Systems, 2012

Contents Glossary 1 Definition and Importance of the Subject 2 1. Introduction 2 2. The volume class 3 3. The fundamental questions 4 4. Lebesgue measure and local properties of volume 4 5. Ergodicity of the basic examples 6 6. Hyperbolic systems 8 7. Beyond uniform hyperbolicity 15 8. The presence of critical points and other singularities 19 9. Future directions 22 References 22 Glossary Conservative, Dissipative: Conservative dynamical systems (on a compact phase space) are those that preserve a finite measure equivalent to volume. Hamiltonian dynamical systems are important examples of conservative systems. Systems that are not conservative are called dissipative. Finding physically meaningful invariant measures for dissipative maps is a central object of study in smooth ergodic theory.

2015

Let T be an aperiodic automorphism of a standard probability space (X, m) and K be either C or R. We prove that the set is dense in L°°(X,SL(2,K)). {A € L°°(XiSL(2,K)) | hmon'l\og(\\A(Tn^1x)"-A(Tx)A(x)\\) > 0, a.e.} We apply this to show that the set of coboundaries are dense in L°°(X, SO(2, R)) or that coboundaries are dense in L°°(X, SU(2)).

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.

Advances in Mathematics, 1985

We show that one-dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an absolutely continuous invariant measure. If f is topologically transitive, some power of f is mixing and, in particular, the correlation of Hölder continuous observables decays to zero. The main objective of this paper is to show that the rate of decay of correlations is determined, in some situations, by the average rate at which typical points start to exhibit exponential growth of the derivative. http://journals.cambridge.org Downloaded: 07 Dec 2012 IP address: 200.128.60.106 638 J. F. Alves et al 1.2. Decay of correlations. Positive Lyapunov exponents are known to be a cause of sensitive dependence on initial conditions and other dynamical features which give rise to a degree of chaoticity or stochasticity in the dynamics. We can formalize this idea through the notion of mixing with respect to some invariant measure. Definition 2. A probability measure µ defined on the Borel sets of I is said to be f-invariant if µ(f −1 (A)) = µ(A) for every Borel set A ⊂ I. Definition 3. A map f is said to be mixing with respect to some f-invariant probability measure µ if |µ(f −n (A) ∩ B) − µ(A)µ(B)| → 0, when n → ∞, for any measurable sets A, B. One interpretation of this property is that the conditional probability of B given f −n (A), i.e. the probability that the event A is a consequence of the event B having occurred at some time in the past, is asymptotically the same as if the two events were completely independent. This is sometimes referred to as a property of loss of memory, and thus in some sense of stochasticity, of the system. A natural question of interest both for application and for intrinsic reasons, therefore, is the speed at which such loss of memory occurs. Standard counterexamples show that, in general, there is no specific rate: it is always possible to choose sets A and B for which mixing is arbitrarily slow. However, this notion can be generalized in the following way. Definition 4. For a map f : I → I preserving a probability measure µ and functions ϕ, ψ ∈ L 1 (µ), we define the correlation function C n = C n (ϕ, ψ) = (ϕ • f n)ψ dµ − ϕ dµ ψ dµ .

GAMM-Mitteilungen, 2009

Key words perturbations, skew product flows, spectra MSC (2000) 37H10, 34D08, 93B05 This paper presents an overview of topological, smooth, and control techniques for dynamical systems and their interrelations for the study of perturbed systems. We concentrate on spectral analysis via linearization of systems. Emphasis is placed on parameter dependent perturbed systems and on a comparison of the Markovian and the dynamical structure of systems with Markov diffusion perturbation process. A number of applications is provided.

Handbook of Dynamical Systems, 2006