Two remarks on the set of recurrent vectors

TL;DR

This paper addresses open questions about the structure of recurrent vectors in linear operators, demonstrating that certain sets can be non-dense and meager, thus providing new insights into their topological and linear properties.

Contribution

It provides counterexamples showing the set of recurrent vectors can be non-dense lineable and meager, resolving two open problems in the field.

Findings

Existence of recurrent operators with non-dense lineable sets of recurrent vectors

Construction of reiteratively recurrent and cyclic operators with meager sets of reiteratively recurrent vectors

Abstract

We solve in the negative two open problems, related to the linear and topological structure of the set of recurrent vectors, asked by Sophie Grivaux, Alfred Peris and the first author of this paper. Firstly, we show that there exist recurrent operators whose set of recurrent vectors is not dense lineable; and secondly, we construct operators which are reiteratively recurrent and cyclic, but whose set of reiteratively recurrent vectors is meager.