The quasi-sure ratio ergodic theorem

A Quasi-ergodic Theorem for Evanescent Processes

We prove a conditioned version of the ergodic theorem for Markov processes, which we call a quasi-ergodic theorem. We also prove a convergence result for conditioned processes as the conditioning event becomes rarer.

On the ratio ergodic theorem for group actions

We study the ratio ergodic theorem (RET) of Hopf for group actions. Under a certain technical condition, if a sequence of sets {Fn} in a group satisfy the RET, then there is a finite set E such that {EFn} satisfies the Besicovitch covering property. Consequently for the abelian group G = ⊕n=1Z there is no sequence Fn ⊆ G along which the RET holds, and in many finitely generated groups, includin...

A Mean Ergodic Theorem For Asymptotically Quasi-Nonexpansive Affine Mappings in Banach Spaces Satisfying Opial's Condition

A Ratio Ergodic Theorem for Multiparameter Non-singular Actions

We prove a ratio ergodic theorem for non-singular free Z and R actions, along balls in an arbitrary norm. Using a Chacon-Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that can concentrate on (thickened) boundaries of balls in R. The proof relies on geometric properties of norms, including the Besicovitch covering lemma and the fact tha...

The Ergodic Theorem

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...