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Jean-François Bertazzon

Publications and source records attributed to Jean-François Bertazzon.

8 recordsLinked to original sources
arXiv2018

Majoration of the dimension of the space of concatenated solutions of a specific pantograph equation

For each $λ\in \mathbb N^*$, we consider the integral equation: \[ \int_{λy} ^{λx} f(t)\, d t = f(x) - f(y) \mbox{ for every $(x,y)\in {\mathbb R}_+^2$,} \] where $f$ is the concatenation of two continuous functions $f_a,f_b:[0,λ] \rightarrow {\mathbb R}$ along a word $u= u_0u_1\cdots\in\{a,b\}^{\mathbb N}$ such that $u=σ(u)$, where $σ$ is a $λ$-uniform substitution satisfying some combinatorial conditions. There exists some non-trivial solutions. We show in this work that the dimension of the set of solutions is at most two.

math.CO↗
arXiv2015

Étude d'une équation intégrale avec des méthodes combinatoires

We build bounded solutions to a linear integral equation. Our functions are built as limit of iterated Birkhoff sums over auto-similar dynamical systems. Nous construisons des solutions bornées à une équation intégrale. Notre méthode repose sur une limite de sommes de Birkhoff itérées au-dessus de certains systèmes dynamiques auto-similaires.

math.DS↗
arXiv2012

Note sur la continuité de la projection dans les facteurs de Host-Kra

B. Host and B. Kra (2005) have introduced the characteristic factors for studying cubic ergodic averages. These factors allow one to resolve, in particular, multiple recurrence problems introduced by H. Furstenberg (1977). We show here that the continuity of the projection of the system in its characteristic factors characterises cubic averages.

math.DS↗
arXiv2012

Fonction complexité associée à une application ergodique du tore

In this article we give the optimal lower bound for the complexity function of a planar translation which induces an ergodic rotation of the torus $\mathbb{R}^2 / \mathbb{Z}^2$. In addition, we give an explicit calculation of this complexity for the application $(x,y) \mapsto (x+1/ϕ^2,y+x-1/ (2ϕ^3))$, where $ϕ$ is the golden mean. Nous proposons dans ce travail de minorer de manière optimale la fonction complexité associée à une translation du plan, qui induit une rotation ergodique du tore $\mathbb R^2/\mathbb Z^2$. De plus, nous donnons un calcul explicite de cette complexité pour l'application $(x,y)\mapsto(x+1/ϕ^2,y+x-1/(2ϕ^3))$, où $ϕ$ désigne le nombre d'or.

math.DS↗
arXiv2012

Resolution of an integral equation with the Thue-Morse sequence

It is a classical fact that the exponential function is solution of the integral equation $ \int_0^X f(x)dx + f(0) =f(X)$. If we slightly modify this equation to $ \int_0^X f(x)dx+f(0)=f(αX)$ with $α\in ]0,1[$, it seems that no classical techniques apply to yields solutions. In this article, we consider the parameter $α=1/2$. We will show the existence of a solution wich takes the values of the Thue-Morse sequence on the odd integers.

math.CO↗
arXiv2011

Symbolic approach and induction in the Heisenberg group

We associate a homomorphism in the Heisenberg group to each hyperbolic unimodular automorphism of the free group on two generators. We show that the first return-time of some flows in "good" sections, are conjugate to niltranslations, which have the property of being self-induced.

math.DS↗
arXiv2010

Convergence uniforme des moyennes ergodiques des fonctions continues

The goal of this work is to study the space of continuous functions whose ergodic averages converge everywhere towards a continuous function. We will connect, as in the case of a metric study, the convergence of the ergodic averages and the projection of continuous functions on the subspace of invariant functions. We will see that this determines the continuity of the projection of the system onto a certain factor.

math.DS↗