Searcharxiv⌕ Search

arXiv subjects

El Houcein El Abdalaoui

Publications and source records attributed to El Houcein El Abdalaoui.

At least 19 recordsLinked to original sources
arXiv2026

Operator ergodic theorems with Möbius "weights"

Motivated by Sarnak's conjecture in topological dynamics for the Möbius function $μ$, we study, for a power-bounded $T$ on a Banach space $E$, the weak convergence $$ (*) \qquad \qquad \frac1N\sum_{n=1}^N μ(n)T^nv \to 0 \text{ weakly } \forall v\in E. $$ For that, we introduce a notion of dynamical entropy for operators, which we denote $h^*_{top}(T)$, and show that if Sarnak's conjecture is true, then $h^*_{top}(T)=0$ implies the desired convergence (*). We conclude an equivalent operator formulation of Sarnak's conjecture. For several classes of operators we prove that (*) holds, and that $h^*_{top}(T)=0$.

math.DS↗
arXiv2019

On the homogeneous ergodic bilinear averages with Möbius and liouville weights

It is shown that the homogeneous ergodic bilinear averages with Möbius or Liouville weight converge almost surely to zero, that is, if $T$ is a map acting on a probability space $(X,\mathcal{A},μ)$, and $a,b \in \mathbb{Z}$, then for any $f,g \in L^2(X)$, for almost all $x \in X$, $${\frac{1}{N}}\sum_{n=1}^{N} ν(n) f(T^{an}x) g(T^{bn}x) \longrightarrow 0, \text{ as } N \longrightarrow +\infty, $$ where $ν$ is the Liouville function or the Möbius function. We further obtain that the convergence almost everywhere holds for the short interval with the help of Zhan's estimation. Also our proof yields a simple proof of Bourgain's double recurrence theorem. Moreover, we establish that if $T$ is weakly mixing and its restriction to its Pinsker algebra has singular spectrum, then for any integer $k \geq 1$, for any $f_j\in L^{\infty}(X)$, $j=1,\cdots,k,$ for almost all $x \in X$, we have $${\frac{1}{N}} \sum_{n=1}^{N} ν(n) \prod_{j=1}^{k}f_j({T_j}^nx) \longrightarrow 0, \text{ as } N \longrightarrow +\infty,$$ where $T_j$ are some powers of $T$, $j=1,\cdots,k$.

math.CA↗
arXiv2018

Disjointness of the Möbius Transformation and Möbius Function

We study the distribution of the sequence of elements of the discrete dynamical system generated by the Möbius transformation $x \mapsto (ax + b)/(cx + d)$ over a finite field of $p$ elements. Motivated by a recent conjecture of P. Sarnak, we obtain nontrivial estimates of exponential sums with such sequences that imply that trajectories of this dynamical system are disjoined with the Möbius function.

math.NT↗
arXiv2018

Möbius disjointness for models of an ergodic system and beyond

Given a topological dynamical system $(X,T)$ and an arithmetic function $\boldsymbol{u}\colon\mathbb{N}\to\mathbb{C}$, we study the strong MOMO property (relatively to $\boldsymbol{u}$) which is a strong version of $\boldsymbol{u}$-disjointness with all observable sequences in $(X,T)$. It is proved that, given an ergodic measure-preserving system $(Z,\mathcal{D},κ,R)$, the strong MOMO property (relatively to $\boldsymbol{u}$) of a uniquely ergodic model $(X,T)$ of $R$ yields all other uniquely ergodic models of $R$ to be $\boldsymbol{u}$-disjoint. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms on nilmanifolds, discrete spectrum automorphisms, systems given by some substitutions of constant length (including the classical Thue-Morse and Rudin-Shapiro substitutions), systems determined by Kakutani sequences are Möbius (and Liouville) disjoint. The validity of Sarnak's conjecture implies the strong MOMO property relatively to $\boldsymbolμ$ in all zero entropy systems, in particular, it makes $\boldsymbolμ$-disjointness uniform. The absence of strong MOMO property in positive entropy systems is discussed and, it is proved that, under the Chowla conjecture, a topological system has the strong MOMO property relatively to the Liouville function if and only if its topological entropy is zero.

math.DS↗
arXiv2016

The Rudin-Shapiro polynomials and The Fekete polynomials are not $L^α$-flat

We establish that the Rudin-Shapiro polynomials are not $L^α$-flat, for any $α\geq 0$. We further prove that the "truncated" Rudin-Shapiro sequence cannot generate a sequence of $L^α$-flat polynomials, for any $α\geq 0$. In the appendix, we present a simple proof of the fact that the Fekete polynomials and the modified or shifted Fekete polynomials are not $L^α$-flat, for any $α\geq 0$.

math.DS↗
arXiv2015

The Chowla and the Sarnak conjectures from ergodic theory point of view

We rephrase the conditions from the Chowla and the Sarnak conjectures in abstract setting, that is, for sequences of numbers in {-1,0,1}, and introduce several natural generalizations. We study the relationships between these properties and other notions from topological dynamics and ergodic theory.

math.DS↗
arXiv2015

Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals

We show that Sarnak's conjecture on Möbius disjointness holds in every uniquely ergodic modelof a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial $P\in\R[x]$ with irrational leading coefficient and for each multiplicative function $\bnu:\N\to\C$, $|\bnu|\leq1$, we have\[ \frac{1}{M} \sum\_{M\le m\textless{}2M} \frac{1}{H} \left| \sum\_{m\le n \textless{} m+H} e^{2πiP(n)}\bnu(n) \right|\longrightarrow 0 \] as $M\to\infty$, $H\to\infty$, $H/M\to 0$.

math.DS↗
arXiv2015

A cubic nonconventional ergodic average with Möbius and Liouville weight

It is shown that the cubic nonconventional ergodic average of order 2 with Möbius and Liouville weight converge almost surely to zero. As a consequence, we obtain that the Cesàro mean of the self-correlations and some moving average of the self-correlations of Möbius and Liouville functions converge to zero.

math.DS↗
arXiv2014

Generalized Riesz Products on the Bohr compactification of $\R$

We study a class of generalized Riesz products connected to the spectral type of some class of rank one flows on $\R$. Applying a Central Limit Theorem of Kac, we exhibit a large class of singular generalized Riesz products on the Bohr compactification of $\R$. Moreover, we discuss the problem of the flat polynomials in this setting.

math.DS↗
arXiv2013

A dynamical point of view on the set of B-free integers

We extend the study of the square-free flow, recently introduced by Sarnak, to the more general context of B-free integers, that is to say integers with no factor in a given family B of pairwise relatively prime integers, the sum of whose reciprocals is finite. Relying on dynamical arguments, we prove in particular that the distribution of patterns in the characteristic function of the B-free integers follows a shift-invariant probability measure, and gives rise to a measurable dynamical system isomorphic to a specific minimal rotation on a compact group. As a by-product, we get the abundance of twin B-free integers. Moreover, we show that the distribution of patterns in small intervals also conforms to the same measure. When elements of B are squares, we introduce a generalization of the Möbius function, and discuss a conjecture of Chowla in this broader context.

math.DS↗
arXiv2013

0-1 sequences of the Thue-Morse type and Sarnak's conjecture

We show that the images via $z\mapsto z^m$ of the continuous part of the spectral measures of the dynamical systems generated by the 0-1 sequences of the Thue-Morse type are pairwise mutually singular for different odd numbers $m\in\N$. Sarnak's conjecture on orthogonality with the Möbius function is shown to hold for such dynamical systems. The same conjecture is shown to hold for all systems induced by regular Toeplitz sequences. A non-regular Toeplitz sequence for which Sarnak's conjecture fails is constructed.

math.DS↗
arXiv2013

On the Fibonacci complex dynamical systems

We consider in this paper a sequence of complex analytic functions constructed by the following procedure $f_n(z)=f_{n-1}(z)f_{n-2}(z)+c$, where $c\in\C$ is a parameter. Our aim is to give a thorough dynamical study of this family, in particular we are able to extend the familiar notions of Julia sets and Green function and to analyze their properties. As a consequence, we extend some well-known results. Finally we study in detail the case where $c$ is small.

math.DS↗
arXiv2013

On spectral disjointness of powers for rank-one transformations and Möbius orthogonality

We study the spectral disjointness of the powers of a rank-one transformation. For a large class of rank-one constructions, including those for which the cutting and stacking parameters are bounded, and other examples such as rigid generalized Chacon's maps and Katok's map, we prove that different positive powers of the transformation are pairwise spectrally disjoint on the continuous part of the spectrum. Our proof involves the existence, in the weak closure of {U_T^k: k in Z}, of "sufficiently many" analytic functions of the operator U_T. Then we apply these disjointness results to prove Sarnak's conjecture for the (possibly non-uniquely ergodic) symbolic models associated to these rank-one constructions: All sequences realized in these models are orthogonal to the Möbius function.

math.DS↗
arXiv2009

On the spectrum of $α$-rigid maps

It is shown that there exists an $α$-rigid transformation with $α$ less or equal to $\frac12$ whose spectrum has Lebesgue component. This answers the question raised by Klemes and Reinhold in \cite{Klemes-Reinhold}. We exhibit also a large class of $α$-rigid transformations with singular spectrum.

math.DS↗
arXiv2009

Approximate transitivity property and Lebesgue spectrum

Exploiting a spectral criterion for a system not to be AT we give some new examples of zero entropy systems without the AT property. Our examples include those with finite spectral multiplicity -- in particular we show that the system arising from the Rudin-Shapiro substitution is not AT. We also show that some nil-rotations on a quotient of the Heisenberg group as well as some (generalized) Gaussian systems are not AT. All known examples of non AT-automorphisms contain a Lebesgue component in the spectrum.

math.DS↗
arXiv2005

Weak mixing of maps with bounded cutting parameter

In the class of Ornstein transformations the mixing property satisfies a 0-1 law. Here we consider Ornstein's construction with bounded cutting parameter. In fact, these latter transformations are not mixing, however it is proved that the weak mixing property occurs with probability one. Our situation is similar to the case of interval exchange transformations whose link with the cutting and stacking construction relies in a dynamical process called Rauzy induction.

math.DS↗