Common hypercyclic vectors for certain families of differential operators

TL;DR

This paper proves the existence of a universal entire function that is hypercyclic for a family of differential operators depending on a sequence and a complex parameter, extending previous results to a large measure set.

Contribution

It establishes the existence of a common hypercyclic vector for a family of differential operators, strengthening prior results and covering a full measure subset of complex numbers.

Findings

Existence of a universal entire function for the family of operators.

Extension of hypercyclicity to a full measure set of complex parameters.

Improved version of previous hypercyclicity results.

Abstract

Let (k(n)) n=1,2,... be a strictly increasing sequence of positive integers . We consider a specific sequence of differential operators Tk(n),{\lambda} , n=1,2,... on the space of entire functions , that depend on the sequence (k(n)) n=1,2,... and the non-zero complex number {\lambda} . We establish the existence of an entire function f , such that for every positive number {\lambda} the set {Tk(n),{\lambda} ,n=1,2,... } is dense in the space of entire functions endowed with the topology of uniform convergence on compact subsets of the complex plane . This provides the best possible strenghthened version of a corresponding result due to Costakis and Sambarino [9] . From this and using a non-trivial result of Weyl which concerns the uniform distribution modulo 1 of certain sequences and Cavalieri principle we can extend our result for a subset of the set of complex numbers with full 2-dimentional Lebesque measure .