A Proof of Liggett's Version of the Subadditive Ergodic Theorem

Journal of Mathematical Analysis and Applications, 1991

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

Ergodic Theory and Dynamical Systems, 2012

We introduce methods that allow us to derive continuous-time versions of various discrete-time ergodic theorems. We then illustrate these methods by giving simple proofs and refinements of some known results as well as establishing new results of interest.

Transactions of the American Mathematical Society, 1998

It is shown that any bounded weight sequence which is good for all probability preserving transformations (a universally good weight) is also a good weight for any L 1 L_{1} -contraction with mean ergodic (ME) modulus, and for any positive contraction of L p L_{p} with 1 &gt; p &gt; ∞ 1 &gt; p &gt;\infty . We extend the return times theorem by proving that if S S is a Dunford-Schwartz operator (not necessarily positive) on a Lebesgue space, then for any g g bounded measurable { S n g ( ω ) } \{S^{n} g(\omega )\} is a universally good weight for a.e. ω . \omega . We prove that if a bounded sequence has &quot;Fourier coefficents&quot;, then its weighted averages for any L 1 L_{1} -contraction with mean ergodic modulus converge in L 1 L_{1} -norm. In order to produce weights, good for weighted ergodic theorems for L 1 L_{1} -contractions with quasi-ME modulus (i.e., so that the modulus has a positive fixed point supported on its conservative part), we show that the modulus of the tenso...

Acta Mathematica, 1994

2003

We prove here that in the Theorem on Local Ergodicity for Semi-Dispersive Billiards (proved by N. I. Chernov and Ya. G. Sinai in 1987) the recently added condition (by P. B\'alint, N. Chernov, D. Sz\'asz, and I. P. T\'oth, in order to save this fundamental result) on the algebraic character of the smooth boundary components of the configuration space is unnecessary. Having saved the theorem in its original form by using additional ideas in the spirit of the initial proof, the result becomes stronger and it applies to a larger family of models.

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.

We develop a general framework for the inverse mean ergodic theorems with rates for operator semigroups thus completing a construction of the theory initiated in and .

Doklady Mathematics, 2007

Given an ergodic semigroup of transformations T t of a probability space (X, A, µ), we introduce and study measures on X induced by mappings s → T ts x from [0, +∞) to X and probability measures on [0, +∞). We obtain several results related the ergodic theorem in the form found recently by Kozlov and Treshchev.

Mathematical Research Letters, 2011

We create a general framework for the study of rates of decay in mean ergodic theorems. As a result, we unify and generalize results due to Assani, Cohen, Cuny, Derriennic, and Lin dealing with rates in mean ergodic theorems in a number of cases. In particular, we prove that the Cesàro means of a power-bounded operator applied to elements from the domain of its abstract one-sided ergodic Hilbert transform decay logarithmically, and this decay is best possible under natural spectral assumptions.