ℱ-Transitivity of Backward Shifts on Lattice Graphs
Abstract
We study ℱ-transitivity of backward shifts on finite- and infinite-width directed lattice graphs. For the unilateral and bilateral finite-width lattices on weighted ℓp- or c0-spaces, we obtain exact weight characterizations of ℱ- and ℱ̃-transitivity for an arbitrary Furstenberg family ℱ. If ℱ is finitely invariant, these conditions also characterize topological ℱ-recurrence. For the infinite quadrant lattice with radial weights on weighted ℓ2-spaces, a finite-dimensional reduction and sharp smallest-singular-value estimates for Pascal-type transfer matrices yield equivalent characterizations of ℱ-transitivity, hypercyclicity and mixing. This provides a partial answer, in the radial ℓ2-setting, to the open problem posed by Baranov, Lishanskii and Papathanasiou concerning hypercyclicity on the infinite lattice graph. Examples at the critical exponential growth rate illustrate the finer dynamical information provided by our criteria.
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BibTeX
@misc{chen2026ℱtransitivity,
title = {{$\mathcal F$-Transitivity of Backward Shifts on Lattice Graphs}},
author = {Xiang Chen and Cui Wang and Ze-hua Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34308},
eprint = {2609.34308},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34308},
}
RIS
TY - UNPB
TI - ℱ-Transitivity of Backward Shifts on Lattice Graphs
AU - Xiang Chen
AU - Cui Wang
AU - Ze-hua Zhou
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34308
UR - https://arxiv.org/abs/2609.34308
AB - We study ℱ-transitivity of backward shifts on finite- and infinite-width directed lattice graphs. For the unilateral and bilateral finite-width lattices on weighted ℓ^p- or c_0-spaces, we obtain exact weight characterizations of ℱ- and ℱ̃-transitivity for an arbitrary Furstenberg family ℱ. If ℱ is finitely invariant, these conditions also characterize topological ℱ-recurrence. For the infinite quadrant lattice with radial weights on weighted ℓ^2-spaces, a finite-dimensional reduction and sharp smallest-singular-value estimates for Pascal-type transfer matrices yield equivalent characterizations of ℱ-transitivity, hypercyclicity and mixing. This provides a partial answer, in the radial ℓ^2-setting, to the open problem posed by Baranov, Lishanskii and Papathanasiou concerning hypercyclicity on the infinite lattice graph. Examples at the critical exponential growth rate illustrate the finer dynamical information provided by our criteria.
ER -