On random almost periodic series and random ergodic theory

The Annals of Probability, 2006

In this paper we study the ergodic theory of a class of symbolic dynamical systems (Ω, T, µ) where T : Ω → Ω the left shift transformation on Ω = ∞ 0 {0, 1} and µ is a σ-finite T -invariant measure having the property that one can find a real number d so that µ(τ d ) = ∞ but µ(τ d−ǫ ) < ∞ for all ǫ > 0, where τ is the first passage time function in the reference state 1. In particular we shall consider invariant measures µ arising from a potential V which is uniformly continuous but not of summable variation. If d > 0 then µ can be normalized to give the unique non-atomic equilibrium probability measure of V for which we compute the (asymptotically) exact mixing rate, of order n −d . We also establish the weak-Bernoulli property and a polynomial cluster property (decay of correlations) for observables of polynomial variation. If instead d ≤ 0 then µ is an infinite measure with scaling rate of order n d . Moreover, the analytic properties of the weighted dynamical zeta function and those of the Fourier transform of correlation functions are shown to be related to one another via the spectral properties of an operator-valued power series which naturally arises from a standard inducing procedure. A detailed control of the singular behaviour of these functions in the vicinity of their non-polar singularity at z = 1 is achieved through an approximation scheme which uses generating functions of a suitable renewal process. In the perspective of differentiable dynamics, these are statements about the unique absolutely continuous invariant measure of a class of piecewise smooth interval maps with an indifferent fixed point.

Israel Journal of Mathematics, 2005

We prove extensions of Menchoff's inequality and the Menchoff-Rademacher theorem for sequences {f n } ⊂ L p , based on the size of the norms of sums of sub-blocks of the first n functions. The results are applied to the study of a.e. convergence of series n a n T n g n α when T is an L 2-contraction, g ∈ L 2 , and {a n } is an appropriate sequence. Given a sequence {f n } ⊂ L p (Ω, µ), 1 < p ≤ 2, of independent centered random variables, we study conditions for the existence of a set of x of µ-probability 1, such that for every contraction T on L 2 (Y, π) and g ∈ L 2 (π), the random power series n f n (x)T n g converges π-a.e. The conditions are used to show that for {f n } centered i.i.d. with f 1 ∈ L log + L, there exists a set of x of full measure such that for every contraction T on L 2 (Y, π) and g ∈ L 2 (π), the random series n fn(x)T n g n converges π-a.e.

Pacific Journal of Mathematics, 1990

We study some classes of totally ergodic functions on locally compact Abelian groups. Among other things, we establish the following result: If R is a locally compact commutative ring, 3ί is the additive group of R, χ is a continuous character of 3$ , and p is the function from 3l n (n e N) into 3% induced by a polynomial of n variables with coefficients in R, then the function χ o p either is a trigonometric polynomial on 3ί n or all of its Fourier-Bohr coefficients with respect to any Banach mean on L°°{^n) vanish.

arXiv (Cornell University), 2017

Let T be a weakly almost periodic (WAP) linear operator on a Banach space X. A sequence of scalars (a n ) n≥1 modulates T on Y ⊂ X if 1 n n k=1 a k T k x converges in norm for every x ∈ Y . We obtain a sufficient condition for (a n ) to modulate every WAP operator on the space of its flight vectors, a necessary and sufficient condition for (weakly) modulating every WAP operator T on the space of its (weakly) stable vectors, and sufficient conditions for modulating every contraction on a Hilbert space on the space of its weakly stable vectors. We study as an example modulation by the modified von Mangoldt function Λ ′ (n) := log n1 P (n) (where P = (p k ) k≥1 is the sequence of primes), and show that, as in the scalar case, convergence of the corresponding modulated averages is equivalent to convergence of the averages along the primes We then prove that for any contraction T on a Hilbert space H and x ∈ H, and also for every invertible T with sup n∈Z T n < ∞ on L r (Ω, µ) (1 < r < ∞) and f ∈ L r , the averages along the primes converge. 1 n n k=1 a k T k x, for every weakly almost periodic T and x ∈ X. Some general results are