A Birkhoff type transitivity theorem for non-separable completely metrizable spaces with applications to Linear Dynamics

TL;DR

This paper extends Birkhoff's transitivity theorem to non-separable completely metrizable spaces and applies it to linear dynamics, showing that certain powers and multiples of transitive operators remain transitive.

Contribution

It proves a new Birkhoff type transitivity theorem for non-separable spaces and generalizes results on hypercyclic operators in complex Fréchet spaces.

Findings

Positive powers of topologically transitive operators are transitive.

Unimodular multiples of transitive operators are also transitive.

The results extend known hypercyclic operator properties.

Abstract

In this note we prove a Birkhoff type transitivity theorem for continuous maps acting on non-separable completely metrizable spaces and we give some applications for dynamics of bounded linear operators acting on complex Fr\'{e}chet spaces. Among them we show that any positive power and any unimodular multiple of a topologically transitive linear operator is topologically transitive, generalizing similar results of S.I. Ansari and F. Le\'{o}n-Saavedra V. M\"{u}ller for hypercyclic operators.