Chaotic polynomials on Banach spaces.
type: Item
instance of: scholarly article
title: Chaotic polynomials on Banach spaces.
zbMATH Open document ID: 1042.46019
DOI: 10.1016/S0022-247X(03)00547-X
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-01-06T00:00:00Z
review text: Let \(X\) be a metric space. A continuous map \(f:X\to X\) is called chaotic in the sense of Auslander and Yorke (AY-chaotic), if \(f\) is topologically transitive and has sensitive dependence on initial conditions. It is called chaotic in the sense of Devaney (D-chaotic), if \(f\) is topologically transitive, the set of periodic points of \(f\) is dense in \(X\), and \(f\) has sensitive dependence on initial conditions, where the last property is redundant. Topological transitivity is equivalent to the existence of a hypercyclic vector \(x\), i.e., Orb\((x) = \{f^nx:n=1,2,\ldots\}\) is dense in \(X\), for certain spaces \(X\) (e.g., separable Fréchet spaces). Many examples of hypercyclic linear operators (= 1-homogeneous polynomials) in Banach spaces are known. However, \textit{N. Bernardes} [Quaest. Math. 21, 311--318 (1998; Zbl 0946.47042)] has shown that there are no hypercyclic \(m\)-homogeneous polynomials of degree \(m > 1\). In this self-contained and very readable paper the author answers the question of R. Aron whether there are hypercyclic polynomials of degree \(m > 1\). The main results are the following. Let \(X=\ell_q\), \(1\leq q<\infty \), \(P:X\to X\) given by \( P((x_i)_i) = (p(x_{i+1}))_i\) where \(p: {\mathbb C} \to {\mathbb C}\) is a polynomial of degree \(m>1\) such that \(p(0)=0\). Then 1. \(P\) is AY-chaotic iff 0 belongs to the Julia set of \(p\), 2. if \(| p(0)| > 1\) then \(P\) is D-chaotic, 3. if \(\ell_q\) is replaced by \(c_0\) then \(P\) is D-chaotic iff 0 belongs to the Julia set of \(p\). It remains an open problem whether \(| p(0)| > 1\) is necessary in 2.
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 47H60
Mathematics Subject Classification ID: 47A16
zbMATH DE Number: 2022071
zbMATH Keywords: hypercyclic vectors
zbMATH Keywords: polynomial operators
zbMATH Keywords: chaos
zbMATH Keywords: Julia sets
author: A. Peris
reviewed by: Hans-Andreas Braunß
MaRDI profile type: Publication
cites work: On Devaney's Definition of Chaos
cites work: ON ORBITS OF POLYNOMIAL MAPS IN BANACH SPACES
cites work: Universal families and hypercyclic operators
cites work: CHAOS FOR BACKWARD SHIFT OPERATORS
cites work: Chaotic polynomials on Fréchet spaces
cites work: On orbits of elements
full work available at URL: s0022-247x(03)00547-x
OpenAlex ID: W2060897400
description: scientific article; zbMATH DE number 2022071