Frequently recurrent backward shifts
Functional Analysis
2026-04-23 v2 Dynamical Systems
Abstract
We study frequently recurrent unilateral and bilateral backward shift operators on Fr\'echet sequence spaces. We prove that if a backward shift admits a non-zero frequently recurrent vector, then it supports a dense set of such vectors, so that the operator is frequently recurrent. As a consequence, we provide two different characterizations for frequently recurrent backward shift operators and we show dense lineability of the set of the set of frequently recurrent vectors.
Cite
@article{arxiv.2407.11799,
title = {Frequently recurrent backward shifts},
author = {Rodrigo Cardeccia and Santiago Muro},
journal= {arXiv preprint arXiv:2407.11799},
year = {2026}
}
Comments
v2: fixed a gap in v1. To address it, we introduce the notion of recurrent arithmetic thickening, which preserves the arithmetic structure and positive lower density. We also add a new section explaining this construction