- Université de Rouen
- Faculté des Sciences et Techniques
- Laboratoire de Mathématiques Raphaël Salem
- UMR 6085 CNRS-Université de Rouen
- Avenue de l'Université, BP.12
- Technopôle du Madrillet
- F76801 Saint-Étienne-du-Rouvray
- Tél. (+ 33) (0)2 32 95 52 27
- Fax (+ 33) (0)2 32 95 52 86
- elhoucein.elabdalaoui@univ-rouen.fr

##### Research Interests

- Ergodic Theory and Dynamical Systems
- Spectral Theory of Dynamical Systems
- Julia sets and complex Analysis
- Probability and Number Theory
- Harmonic Analysis and Fonctionnel Analysis.
- Research Interests In one word by pic

- I just upload on arxiv a new version of my recent work on the Rudin-Shapiro and Fekete polynomials. It is proved that the Rudin-Shapiro and Fekete polynomials. are not L^\alpha-flat, \alpha \geq 0. Please notice that a nice pic is add as a keyword .
- I further upload on arxiv a revised version of my work on the
convergence almost everywhere of the Furstenberg avreage. It is shown
that along a subsequence the multiple ergodic averages of
commuting invertible measure preserving transformations of a Lebesgue
probability space converge almost everywhere. The proof provides a
example of non-singular dynamical system for which the maximal ergodic
inequality does not hold. We further get that the non-singular strategy
to solve the pointwise convergence of the Furstenberg ergodic averages
fails.

- I just upload on arxiv a new version of my recent work on Banach Problem with the alternative proof. This alternative proof has been announced in the August's version. Please notice thats this version start with a nice pic!.
- I
add pic with Professors Y. Sinai, M. Nadkarni, B.
Weiss, I. Assani , W. Gangbo, X. Ye and C. Silva

- Some Results: (*) The Banach
Problem from the Scootich Book on finding a conservative dynamical system with simple Lebesgue
spectrum
has a positive answer in the class
of an infinite measure preserving transformation. Precisely, there
exist a rank-one infinite measure preserving transformationwith simple spectrum.

- (**) There exist a sequence of L^1-flat polynomials with coefficients 0 and 1.
- (***) From this we get a positive answer to Littelwood-Newman problem for L^1-flatness and we answer the L^1 Bourgain’s question.
- (****) We further gives a positive answer to the Mahler problem on the existence of sequence of L^2-normalized Newman polynomials which converge to 1.
- (****) Whence this answers in the positive three questions of long standing in the ergodic theory, harmonic and complex analysis and the number theory (on Arxiv, August 2015).
- Simple Proof of Bourgain theorem on the singularity of the spectrum of Ornstein Class (2 pages on Arxiv, August 2015).
- 1) Non-conventional ergodic avreage with Möbius and Liouville functions and Sarnak's conjecture with application to the study of the self-correlation of the Möbius and Liouvile functions (work in progress, el Abdalaoui-XiangDong Ye, 2015, preliminary version on Arxiv).
- 2) Sarnak's conjecture for uniquely ergodic models of rotations (work in progress, el Abdalaoui-Lemanczyk-delaRue, 2015, on Arxiv).
- 1) There exist a non-singular map which has a simple Lebesgue component in the spectrum (el Abdalaoui-Nadkarni, 2014).
- 2) In the class of rank one maps three notorious problems are equivalents : Banach problem, Littewood problem and Mahler problem!. (el Abdalaoui-Nadkarni, 2013)
- 3) Kahane ultraflat polynomials give rise to Generalized Riesz product equivalent to Lebesgue measure (el Abdalaoui-Nadkarni, 2013).

*) A Class of Littewood polynomials that are not L^\alpha-flats (soon on Arxiv).

**) A cubic nonconventional ergodic average with M\"{o}bius and Liouville weight converge almost everywhere (soon on Arxiv).

***) I will put soon on my page a short paper on the real history of LMRS and our departement of mathematics. A history since 1966 in which every one who contribute to this history will be mentionned at exact level of his contribution. The real history of LMRS correspond exactly to the following David Hilbert's Quote :

"Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country."

**) A cubic nonconventional ergodic average with M\"{o}bius and Liouville weight converge almost everywhere (soon on Arxiv).

***) I will put soon on my page a short paper on the real history of LMRS and our departement of mathematics. A history since 1966 in which every one who contribute to this history will be mentionned at exact level of his contribution. The real history of LMRS correspond exactly to the following David Hilbert's Quote :

"Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country."