$\mathcal F$-Transitivity of Backward Shifts on Lattice Graphs

Abstract

We study $\mathcal F$-transitivity of backward shifts on finite- and infinite-width directed lattice graphs. For the unilateral and bilateral finite-width lattices on weighted $\ell^p$- or $c_0$-spaces, we obtain exact weight characterizations of $\mathcal F$- and $\widetilde{\mathcal F}$-transitivity for an arbitrary Furstenberg family $\mathcal F$. If $\mathcal F$ is finitely invariant, these conditions also characterize topological $\mathcal F$-recurrence. For the infinite quadrant lattice with radial weights on weighted $\ell^2$-spaces, a finite-dimensional reduction and sharp smallest-singular-value estimates for Pascal-type transfer matrices yield equivalent characterizations of $\mathcal F$-transitivity, hypercyclicity and mixing. This provides a partial answer, in the radial $\ell^2$-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.