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 ...

in this paper, we introduce the concepts of $2$-isometry, collinearity, $2$%-lipschitz mapping in $2$-fuzzy $2$-normed linear spaces. also, we give anew generalization of the mazur-ulam theorem when $x$ is a $2$-fuzzy $2$%-normed linear space or $im (x)$ is a fuzzy $2$-normed linear space, thatis, the mazur-ulam theorem holds, when the $2$-isometry mapped to a $2$%-fuzzy $2$-normed linear space...

Using the ratio ergodic theorem for a measure preserving transformation in a σ-finite measure space we give a straightforward proof of Derriennic’s reverse maximal inequality for the supremum of ergodic ratios.

In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of R on L(X)-spaces are convergent for d ≥ 3 and p > d d− 1 . This is done by adapting the proof of the spherical maximal theorem by Rubio de Francia so as to obtain directly the ergodic theorem.

We prove that each invariant measure in a non-uniformly hyperbolic system can be approximated by atomic measures on hyperbolic periodic orbits. This contributes to our main result that the mean angle (Definition 1.10), independence number (Definition 1.6) and Oseledec splitting for an ergodic hyperbolic measure with simple spectrum can be approximated by those for atomic measures on hyperbolic ...

We consider infinite measure-preserving non-primitive selfsimilar tiling systems in Euclidean space R. We establish the secondorder ergodic theorem for such systems. The speed of convergence is determined by the Hausdorff dimension of a graph-directed set associated to the substitution rule.

In a recent work [3], the authors established new results about general linear Mahler systems in several variables from perspective of transcendental number theory, such as multivariate extension Nishioka’s theorem. Working with functions and different transformations leads to complications, including need prove vanishing theorem use tools ergodic Ramsey theory Diophantine approximation (e.g., ...

The aim of this thesis is to formulate and prove quantum extensions of the famous Shannon-McMillan theorem and its stronger version due to Breiman. In ergodic theory the Shannon-McMillan-Breiman theorem is one of the fundamental limit theorems for classical discrete dynamical systems. It can be interpreted as a special case of the individual ergodic theorem. In this work, we consider spin latti...

We prove absence of absolutely continuous spectrum for discrete one-dimensional Schrödinger operators on the whole line with certain ergodic potentials, Vω(n) = f(T n(ω)), where T is an ergodic transformation acting on a space Ω and f : Ω → R. The key hypothesis, however, is that f is discontinuous. In particular, we are able to settle a conjecture of Aubry and Jitomirskaya–Mandel’shtam regardi...