Chaotic polynomials on Banach spaces.
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.
scientific article; zbMATH DE number 2022071