2012/05/16 by Felix Pogorzelski, Pogorzelski, Felix, Fabian Schwarzenberger +1
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.DS #math.MP #math.SP
paper · pdf · doi:10.48550/arxiv.1205.3649
40 pages, Sections 8 and 9 of the previous version removed, slight modifications
arxiv created 2013/07/30 · arxiv updated 2013/07/31
In this paper we study unimodular amenable groups. The first part is devoted to results on the existence of uniform families of epsilon-quasi tilings for these groups. In this context, constructions of Ornstein and Weiss are extended by quantitative estimates for the covering properties of the corresponding decompositions. Afterwards, we apply the developed methods to obtain an abstract ergodic theorem for a class of functions mapping subsets of a countable, amenable group into some Banach space. This significantly extends and complements the previous results of Lenz, Müller, Schwarzenberger and Veselić. Further, using the Lindenstrauss ergodic theorem, we describe a link of our results to classical ergodic theory. We conclude with two important applications: the uniform approximation of the integrated density of states on amenable Cayley graphs, as well as the almost-sure convergence of cluster densities in an amenable bond percolation model.