A pointwise ergodic theorem for quasi-pmp graphs
Abstract: We prove a pointwise ergodic theorem for quasi-pmp locally countable graphs, which states that the global condition of ergodicity amounts to locally approximating the means of $L^1$-functions via increasing subgraphs with finite connected components. The pmp version of this theorem was first proven by R. Tucker-Drob using probabilistic methods. Our proof is different: it is constructive and applies more generally to quasi-pmp graphs. It involves introducing a graph invariant, a packedness condition for finite Borel subequivalence relations, and an easy method of exploiting nonamenability. The quasi-pmp setting additionally requires a new gadget for analyzing the interplay between the underlying cocycle and the graph.
Tuesday February 27, 2018 at 3:30 PM in SEO 427