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Journal of Function Spaces and Applications
Volume 2013, Article ID 320961, 6 pages
http://dx.doi.org/10.1155/2013/320961
Research Article

On the Existence of Polynomials with Chaotic Behaviour

1Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, CP 68530, 21945-970 Rio de Janeiro, RJ, Brazil
2IUMPA, Universitat Politècnica de València, Departament de Matemática Aplicada, Edifici 7A, 46022 València, Spain

Received 29 May 2013; Accepted 7 August 2013

Academic Editor: Ajda Fošner

Copyright © 2013 Nilson C. Bernardes Jr. and Alfredo Peris. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We establish a general result on the existence of hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials on locally convex spaces. As a consequence we prove that every (real or complex) infinite-dimensional separable Frèchet space admits mixing (hence hypercyclic) polynomials of arbitrary positive degree. Moreover, every complex infinite-dimensional separable Banach space with an unconditional Schauder decomposition and every complex Frèchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree. We also study distributional chaos for polynomials and show that every infinite-dimensional separable Banach space supports polynomials of arbitrary positive degree that have a dense distributionally scrambled linear manifold.