This paper is concerned with the concept of linear repetitivity in the theory of tilings. We prove a general uniform subadditive ergodic theorem for linearly repetitive tilings. This theorem unifies and extends various known (sub)additive ergodic theorems on tilings. The results of this paper can be applied in the study of both random operators and lattice gas models on tilings.

We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem. Let (X,B, μ) be a probability space, T : X → X a (possibly noninvertible) measurepreserving transformation, and f ∈ L(X,B, μ). Let

a random walk on a lattice is one of the most fundamental models in probability theory. when the random walk is inhomogenous and its inhomogeniety comes from an ergodic stationary process, the walk is called a random walk in a random environment (rwre). the basic questions such as the law of large numbers (lln), the central limit theorem (clt), and the large deviation principle (ldp) are no...

By exploiting the Denjoy theorem in topological dynamics and the unique ergodic theorem in ergodic theory, we will give a classification of all solutions of asymmetric p-Laplacian oscillators with periodic coefficients. AMS (MOS) Subject Classification. Primary: 34D08; Secondary: 37E10, 37A25.

This is an earlier, but more general, version of ”An L Ergodic Theorem for Sparse Random Subsequences”. We prove an L ergodic theorem for averages defined by independent random selector variables, in a setting of general measure-preserving group actions. A far more readable version of this paper is in the works.

We study a stability property of probability laws with respect to small violations of algorithmic randomness. A sufficient condition of stability is presented in terms of Schnorr tests of algorithmic randomness. Most probability laws, like the strong law of large numbers, the law of iterated logarithm, and even Birkhoff’s pointwise ergodic theorem for ergodic transformations, are stable in this...

We offer a proof of the following nonconventional ergodic theorem: Theorem. If Ti : Z y (X,Σ, μ) for i = 1, 2, . . . , d are commuting probability-preserving Z-actions, (IN )N≥1 is a Følner sequence of subsets of Z, (aN )N≥1 is a base-point sequence in Z and f1, f2, . . . , fd ∈ L∞(μ) then the nonconventional ergodic averages

It is observed that a wellnigh trivial application of the ergodic theorem from [3] yields a strong LLN for arbitrary concave moments. Not for publication: we found that Aaronson–Weiss essentially proved Theorem 1, see J. Aaronson, An introduction to infinite ergodic theory (AMS Math. Surv. Mon. 50, 1997), pages 65–66.