Growth of hypercyclic entire functions for some non-convolution operators

Abstract

A continuous linear operator T defined on a Frechet space X is said to be hypercyclic if there exists f ∈ χsuch that, the orbit {Tnf} is dense in X. In this article, we consider the operators introduced by Aron and Markose, defined on the space of entire functions by Tλb f(z) = f(λz+b) λ ∈ ℂ\{1}, b ∈ ℂ, and we aimed to explore the rate of growth of hypercyclic vectors for Tλb. We discover that Tλb, is a weighted backward shift with respect to some basis and this fact allows us to find sharp estimates of the growth of Tλb -hypercyclic vectors. When λ > 1, the Tλb -hypercyclic function growth is similar to the D-hypercyclic functions (D is the differentiation operator), and when λ > 1, theλb -hypercyclic functions can grow very slowly but not arbitrarily slowly. A lower bound of this growth is found in terms of the W-Lambert function. Finally, partial results are obtained for Tλb -frequently hypercyclic functions.