Rate of growth of random analytic functions, with an application to linear dynamics

TL;DR

This paper improves classical inequalities for random analytic functions and applies these results to determine growth rates of functions that are frequently hypercyclic in linear dynamics, especially for chaotic weighted backward shifts.

Contribution

It introduces improved Wiman-Valiron type inequalities for random entire and analytic functions and applies these to analyze growth rates in linear dynamics for hypercyclic functions.

Findings

Enhanced inequalities for random analytic functions

Established growth rates for frequently hypercyclic functions

Applied results to chaotic weighted backward shifts

Abstract

We obtain Wiman-Valiron type inequalities for random entire functions and for random analytic functions on the unit disk that improve a classical result of Erd\H{o}s and R\'enyi and recent results of Kuryliak and Skaskiv. Our results are then applied to linear dynamics: we obtain rates of growth, outside some exceptional set, for analytic functions that are frequently hypercyclic for an arbitrary chaotic weighted backward shift.