Chaotic polynomials on Fréchet spaces

Abstract

Contrary to the case of polynomials on Banach spaces, in which it is known that no hypercyclic homogeneous polynomial of degree m > 2 exists on any Banach space, we construct for each m > 2 a chaotic m-homogeneous polynomial P on the Frechet space '((C). A map T: X -> X on a metric space X is chaotic (see [3]) if (a) T is transitive (which for complete separable X is equivalent to the existence of x E X whose orbit Orb(T, x) := {x, Tx, T2x,... } is dense in X), (b) the periodic points of T are dense in X, and (c) T has sensitive dependence on initial conditions. Banks et al. [1] showed that (c) is redundant in the definition of Devaney. We present an example of a chaotic m-homogeneous continuous polynomial P : H(C) -> t((C). This must be compared with a result of Bernardes [2], who showed that for m > 1 there are no continuous m-homogeneous polynomials admitting a vector with dense orbit (hypercyclic in the usual terminology) on any Banach space. The result of Bernardes is a consequence of the inequality II Pnx II 1) on a Banach space X, since no hypercyclic vector for P could lie on the ball centered at 0 of radius r := 1/ II P II. This implies that P does not admit any hypercyclic vector. On 7H(C) we consider the increasing sequence of norms (11 * IIk)k defined by If((0)MI If llk:= sup k3, keN, fe (C), j_o j! which define the natural Frechet topology on 7-(C). Our notation is standard. We refer to the monograph [6] for Frechet spaces and to [4] for polynomials on locally convex spaces. Theorem 1. For each m natural (m > 2) there exists a chaotic m-homogeneous polynomial P: 7-(C) -(C). (f(j+l) (o))m Proof. Let us define P :H-(C) H((C) by (Pf)(z):= ( )) z3 for every j>0 f E H((C) and for every z E C. P is obviously a well-defined m-homogeneous polynomial. We first prove that P is hypercyclic. Received by the editors December 29, 1997 and, in revised form, February 17, 1998. 1991 Mathematics Subject Classification. Primary 46G20, 46A04, 58F08. This research was supported in part by DGICYT under Proyecto PB94-0541. ?1999 American Mathematical Society 3601 This content downloaded from 157.55.39.212 on Wed, 08 Jun 2016 06:41:03 UTC All use subject to http://about.jstor.org/terms