Ergodic Theory and Dynamical Systems



Chaos for functions of discrete and continuous weighted shift operators


R. DELAUBENFELS a1 and H. EMAMIRAD a2
a1 Scientia Research Institute, 22000 Jordan Run Road, Guysville, Ohio 45735, USA (e-mail: scientia@frognet.net)
a2 Université de Poitiers, Laboratoire de Modélisation Mécanique et de Mathématiques Appliquées, SP2MI, BP 30179, F 86962 Futuroscope Cedex, France (e-mail: emamirad@l3ma.univ-poitiers.fr)

Abstract

For L equal to the unilateral or bilateral shift on a weighted sequence space, we characterize, in terms of f and the weight function, those f holomorphic on the spectrum of L for which f(L) is a chaotic operator. For B equal to d/dx, the generator of left translation, on weighted L^p spaces on [0, \infty) or \mathbb{R}, we similarly characterize those polynomials Q for which the differential operator Q(B) generates a chaotic semigroup.

(Received January 5 2000)
(Revised September 20 2000)