Uniqueness of mean maximizers via an ergodic theorem

On the Mean Ergodic Theorem for Subsequences

With these assumptions we have T defined for every integer n as a 1-1, onto, bimeasurable transformation. Henceforth we shall assume that every set considered is measurable, i.e. an element of a. We shall say that P is invariant if P(A) =P(TA) for every set A, P is ergodic if P is invariant and if P(U^L_oo TA) = 1 for every set A for which P(A) > 0 , and finally P is strongly mixing if P is inv...

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...

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

Application of the Mean Ergodic Theorem to Certain Semigroups

We study the asymptotic behaviour of solutions of the Cauchy problem u′ = (∑n j=1(Aj + A −1 j ) − 2nI ) u, u(0) = x as t → ∞, for invertible isometries A1, . . . , An.

A quantitative Mean Ergodic Theorem for uniformly convex Banach spaces

We provide an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces. Our result can also be viewed as a finitary version in the sense of T. Tao of the Mean Ergodic Theorem for such spaces and so generalizes similar results obtained for Hilbert spaces by Avigad, Gerhardy and Towsner [1] and T. Tao [11].