Alfredo Peris

CHAOS FOR BACKWARD SHIFT OPERATORS

Backward shift operators provide a general class of linear dynamical systems on infinite dimensional spaces. Despite linearity, chaos is a phenomenon that occurs within this context. In this paper we give characterizations for chaos in the sense of Auslander and Yorke [1980] and in the sense of Devaney [1989] of weighted backward shift operators and perturbations of the identity by backward shifts on a wide class of sequence spaces. We cover and unify a rich variety of known examples in different branches of applied mathematics. Moreover, we give new examples of chaotic backward shift operators. In particular we prove that the differential operator I + D is Auslander–Yorke chaotic on the most usual spaces of analytic functions.

On hypercyclicity and supercyclicity criteria

We show that the Hypercyclicity Criterion coincides with other existing hypercyclicity criteria and prove that a wide class of hypercyclic operators satisfy the Criterion. The results obtained extend or improve earlier work of several authors. We also unify the different versions of the Supercyclicity Criterion and show that operators with dense generalised kernel and dense range are supercyclic.

Multi-hypercyclic operators are hypercyclic

Abstract. Herrero conjectured in 1991 that every multi-hypercyclic (respectively, multi-supercyclic) operator on a Hilbert space is in fact hypercyclic (respectively, supercyclic). In this article we settle this conjecture in the affirmative even for continuous linear operators defined on arbitrary locally convex spaces. More precisely, we show that, if

Transitive and Hypercyclic Operators on Locally Convex Spaces

T:E \rightarrow E

Linear Chaos on Fréchet Spaces

is a continuous linear operator on a locally convex space E such that there is a finite collection of orbits of T satisfying that each element in E can be arbitrarily approximated by a vector of one of these orbits, then there is a single orbit dense in E. We also prove the corresponding result for a weaker notion of approximation, called supercyclicity .

Universality and chaos for tensor products of operators

Solutions are provided to several questions concerning topologically transitive and hypercyclic continuous linear operators on Hausdorff locally convex spaces that are not Frechet spaces. Among others, the following results are presented. (1) There exist transitive operators on the space ϕ of all finite sequences endowed with the finest locally convex topology (it was already known that there is no hypercyclic operator on ϕ. (2) The space of all test functions for distributions, which is also a complete direct sum of Frechet spaces, admits hypercyclic operators. (3) Every separable infinite-dimensional Frechet space contains a dense hyperplane that admits no transitive operator. 2000 Mathematics Subject Classification 47A16 (primary), 46A03, 46A04, 46A13, 37D45 (secondary).

Strong mixing measures for linear operators and frequent hypercyclicity

This is a survey on recent results about hypercyclicity and chaos of continuous linear operators between complete metrizable locally convex spaces. The emphasis is put on certain contributions from the authors, and related theorems.

Weakly mixing operators on topological vector spaces

We give sufficient conditions for the universality of tensor products {Tn, ~ Rn : n ∈ N} of sequences of operators defined on Frechet spaces. In particular we study when the tensor product T ~ R of two operators is chaotic in the sense of Devaney. Applications are given for natural operators on function spaces of several variables, in Infinite Holomorphy, and for multiplication operators on the algebra L(E) following the study of Kit Chan.

CHAOTIC ASYMPTOTIC BEHAVIOR OF THE HYPERBOLIC HEAT TRANSFER EQUATION SOLUTIONS

Abstract We construct strongly mixing invariant measures with full support for operators on F -spaces which satisfy the Frequent Hypercyclicity Criterion. For unilateral backward shifts on sequence spaces, a slight modification shows that one can even obtain exact invariant measures.

Chaotic polynomials on Banach spaces

In this paper we review some known characterizations of the weak mixing property for operators on topological vector spaces, extend some of them, and obtain new ones.ResumenEn este artículo revisamos algunas caracterizaciones conocidas de la propiedad débil mezclante para operadores en espacios vectoriales topológicos, extendemos alguna de ellas, y obtenemos otras nuevas.