A Banach space-valued ergodic theorem for amenable groups and applications (Q518555)

The authors study geometric and spectral approximation results for amenable groups. First, they extend the celebrated \(\varepsilon\)-quasi results for amenable groups of Ornstein and Weiss. They give precise effective estimates on the covering and uniformity properties of the tilings under consideration. Second, using the celebrated tiling techniques, they prove an almost-additive ergodic theorem (Theorem 5.5), which is valid for all countable amenable groups. This generalizes the results in the literature. Further, using the Lindenstrauss ergodic theorem, they link their results to classical ergodic theory. Third, they conclude with two important applications: the uniform approximation of the integrated density of states on amenable Cayley graphs and the almost-sure convergence of cluster densities in an amenable bond percolation model.