Idris Assani

Wiener Wintner ergodic theorems

The Mean and Pointwise Ergodic Theorems Wiener Wintner Pointwise Ergodic Theorems Universal Weights for Dynamical Systems J Bourgains Return Times Theorem Extensions of the Return Times Theorem Speed of Convergence in the Uniform Wiener Wintner Theorem Weak Wiener Wintner Dynamical Systems Polynomial Wiener Wintner Ergodic Theorem Extension to More General Operators.

Multiple recurrence and almost sure convergence for weakly mixing dynamical systems

AbstractWe prove the following: Let (X, β, μ,T) be a weakly mixing dynamical system such that the restriction ofT to its Pinsker algebra has singular spectrum, then for all positive integersH, for allfi ∈L∞, 1≤i≤H, the averages

A weighted pointwise ergodic theorem

\frac{1}{N}\sum\limits_{n = 1}^N {f_1 (T^n x)f_2 (T^{2n} x) \cdot \cdot \cdot f_H (T^{Hn} x)} converge a.e. to \prod\limits_{i = 1}^H {\int {f_i d\mu } }

Extension of Wiener-Wintner double recurrence theorem to polynomials

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A good universal weight for nonconventional ergodic averages in norm

We give a negative solution to the following counting problem for measure preserving transformations. Forf∈L+1(μ), is it true that supn (Nn(f)(x)/n) <∞, μ a.e., where Nn(f)(x)=≠{k:f(Tkx)/k>1/n}? One of the consequences is the nonvalidity of J. Bourgain’s Return Time Theorem for pairs of (L1,L1) functions.

Multiple return times theorems for weakly mixing systems

Abstract We prove the following weighted ergodic theorem: Let ( X n ) be an i.i.d. sequence of symmetric random variables such that E (| X 1 | p ) p , 1 p Ω ˜ such that for θ ∈ Ω ˜ the following holds: For all dynamical systems ( Y , G , ν, S ), for all r , 1 r ≤ ∞ and g ∈ L r ( ν ) the averages 1 N ∑ n = 1 N X n ( ω ) g ( S n y ) converge a.e. ν .

The helical transform as a connection between ergodic theory and harmonic analysis

This paper is an update and extension of a result the authors first proved in 2003. The goal of this paper is to study factors which are known to be Lcharacteristic for certain nonconventional averages and prove that these factors are pointwise characteristic for the multiterm return times averages. In memory of Dan Rudolph. Pointwise Characteristic Factors 1

A Wiener—Wintner property for the helical transform

We extend our result on the convergence of double recurrence Wiener-Wintner averages to the case of a polynomial exponent. We show that there exists a unique set of full measure for which the averages