Dynamics of Weighted Backward Shifts on Cesàro Spaces of Rooted Trees

We study the dynamics of weighted backward shifts on Cesaro spaces associated with leafless locally finite rooted trees. We first characterize their boundedness in terms of adjacent level cardinalities and edge weights. We then characterize their $\mathcal{F}$-transitivity by a growth condition involving level cardinalities, products of weights along paths, and a level-dependent Cesaro factor. As consequences, we obtain criteria for hypercyclicity, weak mixing, topological ergodicity, and topological mixing. We also characterize the existence of nonzero orbit limit points and chaotic weighted shifts, the latter in terms of normalized fixed points and unit flows satisfying an explicit summability condition. Examples show that ${\mathcal{F}}_{\underset{‾}{d}>0}$-transitivity need not imply frequent hypercyclicity, that a nonhypercyclic weighted shift may nevertheless have a nonzero orbit limit point, and that topological mixing need not imply chaos.