We consider Delone sets with finite local complexity. We characterize validity of a subadditive ergodic theorem by uniform positivity of certain weights. The latter can be considered to be an averaged version of linear repetitivity. In fact, it is shown that linear repetitivity is equivalent to positivity of weights combined with a certain repulsion property of patterns and a balancedness of th...

Measure-preserving systems arise in a variety of contexts, such as probability theory, information theory, and of course in the study of dynamical systems. However, ergodic theory originated from statistical mechanics. In this setting, T represents the evolution of the system through time. Given a measurable function f : X → R, the series of values f(x), f(Tx), f(T x)... are the values of a phy...

We prove a non conventional pointwise convergence theorem for a nilsystem, and give an explicit formula for the limit.