Non-existence of common hypercyclic entire functions for certain families of translation operators

TL;DR

This paper proves that certain families of translation operators with exponentially growing translates do not share any hypercyclic entire functions, highlighting limitations in their common dynamic behavior.

Contribution

It establishes the non-existence of common hypercyclic entire functions for specific exponentially growing translation operator families, advancing understanding of operator dynamics.

Findings

Families with exponential growth in translation do not have common hypercyclic functions

The result is nearly optimal, indicating tight bounds on such operator families

Provides insight into the limitations of hypercyclicity in translation operators

Abstract

We show that families of translation operators, where the translates grow exponentially fast, do not admit common hypercyclic functions. The result is close to be optimal.