The Ergodic Theorem

Individual ergodic theorem for intuitionistic fuzzy observables using intuitionistic fuzzy state

The classical ergodic theory hasbeen built on σ-algebras. Later the Individual ergodictheorem was studied on more general structures like MV-algebrasand quantum structures. The aim of this paper is to formulate theIndividual ergodic theorem for intuitionistic fuzzy observablesusing  m-almost everywhere convergence, where  m...

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

Non-linear ergodic theorems in complete non-positive curvature metric spaces

Hadamard (or complete $CAT(0)$) spaces are complete, non-positive curvature, metric spaces. Here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. Our results extend the standard non-linear ergodic theorems for non-expansive maps on real Hilbert spaces, to non-expansive maps on Ha...

ON LOCAL HUDETZ g-ENTROPY

In this paper, a local approach to the concept of Hudetz $g$-entropy is presented. The introduced concept is stated in terms of Hudetz $g$-entropy. This representation is based on the concept of $g$-ergodic decomposition which is a result of the Choquet's representation Theorem for compact convex metrizable subsets of locally convex spaces.

Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem

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

Szemerédi’s Theorem via Ergodic Theory

This essay investigates Furstenberg’s proof of Szemerédi’s Theorem. The necessary concepts and results from ergodic theory are introduced, including the Poincaré and Mean Ergodic Theorems which are proved in full. The Ergodic Decomposition Theorem is also discussed. Furstenberg’s Multiple Recurrence Theorem is then stated and shown to imply Szemerédi’s Theorem. The remainder of the essay concen...