El Houcein El Abdalaoui

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).