arXiv subjects

Ali Messaoudi

Publications and source records attributed to Ali Messaoudi.

13 recordsLinked to original sources

arXiv2026

On Generalized Hyperbolicity, Stability, and Shadowing for Linear Operators

This work studies the relations between shadowing, topological stability, Lipschitz structural stability, and pseudo-hyperbolicity for linear dynamical systems on Banach spaces. The main result establishes that pseudo-hyperbolicity implies both topological stability and strong Lipschitz structural stability, whereas strong Lipschitz structural stability implies the shadowing property. In addition, a spectral characterization of pseudo-hyperbolicity is obtained, yielding an equivalence between pseudo-hyperbolicity, topological stability, strong Lipschitz structural stability, and the shadowing property for invertible operators whose eigenspaces associated with unimodular eigenvalues admit closed complements. In particular, these four notions are equivalent on Hilbert spaces. Combined with a recent result of Dragičević and Pituk, these results yield, for a large class of Banach spaces, an invertible operator that has the shadowing property but is not generalized hyperbolic, thereby answering an open problem in linear dynamics.

math.DS↗

arXiv2025

Operators of stochastic adding machines and Julia sets

A stochastic adding machine is a Markov chain on the set of non-negative integers $\mathbb{Z}_{+}$ that models the process of adding one by successively updating the digits of a number's expansion in a given numeration system. At each step, random failures may occur, interrupting the procedure and preventing it from continuing beyond a certain point. The first model of such a stochastic adding machine, constructed for the binary base, was introduced by Killeen and Taylor. Their work was motivated by applications to biological clocks, aiming to model phenomena related to time discrimination and/of psychological judgment. From a mathematical perspective, they characterized the spectrum of the associated transition operator in terms of a filled Julia set. In this paper, we consider a stochastic adding machine based on a bounded Cantor numeration system and extend its definition to a continuous state space--namely, the closure of $\mathbb{Z}_+$ with respect to the topology induced by the Cantor numeration system. This stochastic process naturally induces a transition operator $S$ acting on the Banach space of continuous complex-valued functions over the continuous state space, as well as a fibered filled Julia set $\mathcal{E}$. Our main result describes the spectrum of $S$ in terms of the fibered filled Julia set $\mathcal{E}$. Specifically, if the stochastic adding machines halts with probability one after a finite number of steps, then the spectrum of $S$ coincides with $\mathcal{E}$; otherwise, the spectrum coincides with the boundary $\partial \mathcal{E}$.

math.DS↗

arXiv2021

Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism

We show the existence of invariant ergodic $σ$-additive probability measures with full support on $X$ for a class of linear operators $L: X \to X$, where $L$ is a weighted shift operator and $X$ either is the Banach space $c_0(\mathbb{R})$ or $l^p(\mathbb{R})$ for $1\leq p<\infty$. In order to do so, we adapt ideas from Thermodynamic Formalism as follows. For a given bounded Hölder continuous potential $A:X \to \mathbb{R}$, we define a transfer operator $\mathcal{L}_A$ which acts on continuous functions on $X$ and prove that this operator satisfies a Ruelle-Perron-Frobenius theorem. That is, we show the existence of an eigenfunction for $\mathcal{L}_A$ which provides us with a normalized potential $\overline{A}$ and an action of the dual operator $\mathcal{L}_{\overline{A}}^*$ on the $1$-Wasserstein space of probabilities on $X$ with a unique fixed point, to which we refer to as Gibbs probability. It is worth noting that the definition of $\mathcal{L}_A$ requires an {\it a priori} probability on the kernel of $L$. These results are extended to a wide class of operators with a non-trivial kernel defined on separable Banach spaces.

math.DS↗

arXiv2020

A generalized Grobman-Hartman theorem

We prove that any generalized hyperbolic operator on any Banach space is structurally stable. As a consequence, we obtain a generalization of the classical Grobman-Hartman theorem.

math.DS↗

arXiv2020

Shadowing and Stability in p-adic dynamics

In this paper, we study dynamical properties as shadowing and structural stability for a class of dynamics on $\mathbb{Z}_p$ and $\mathbb{Q}_p$, where $p \geq 2$ is a prime number. In particular, we prove that if $f: \mathbb{Z}_p \to \mathbb{Z}_p$ is a $(p^{-k},p^{m})$ ( $0 < m \leq k$ integers ) locally scaling map then $f$ is shadowing and structurally stable. We also study the number of conjugacy classes of these maps and we consider the above properties for $1$-Lipschitz maps of $\mathbb{Z}_p$ and for extensions of the shift map, contractions and dilatations on $\mathbb{Q}_p$.

math.DS↗

arXiv2019

Stochastic adding machines based on Bratteli diagrams

In this paper, we define some Markov Chains associated to Vershik maps on Bratteli diagrams. We study probabilistic and spectral properties of their transition operators and we prove that the spectra of these operators are connected to Julia sets in higher dimensions. We also study topological properties of these spectra.

math.DS↗

arXiv2019

Shadowing and structural stability in linear dynamical systems

A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by P. Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by J. Palis and C. Pugh around 1968. We will exhibit examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expansive and has the shadowing property. Moreover, we will show that if a structurally stable operator is expansive, then it must be uniformly expansive. Finally, we will characterize the weighted shifts on the spaces $c_0(\mathbb{Z})$ and $\ell_p(\mathbb{Z})$ ($1 \leq p < \infty$) that satisfy the shadowing property.

math.DS↗

arXiv2017

Dynamical properties of random walks

In this paper, we study dynamical properties as hypercyclicity, supercyclicity, frequent hypercyclicity and chaoticity for transition operators associated to countable irreductible Markov chains. As particular cases, we consider simple random walks on Z and Z+.

math.DS↗

arXiv2016

Expansivity and Shadowing in Linear Dynamics

In the early 1970's Eisenberg and Hedlund investigated relationships between expansivity and spectrum of operators on Banach spaces. In this paper we establish relationships between notions of expansivity and hypercyclicity, supercyclicity, Li-Yorke chaos and shadowing. In the case that the Banach space is $c_0$ or $\ell_p$ ($1 \leq p < \infty$), we give complete characterizations of weighted shifts which satisfy various notions of expansivity. We also establish new relationships between notions of expansivity and spectrum. Moreover, we study various notions of shadowing for operators on Banach spaces. In particular, we solve a basic problem in linear dynamics by proving the existence of nonhyperbolic invertible operators with the shadowing property. This also contrasts with the expected results for nonlinear dynamics on compact manifolds, illuminating the richness of dynamics of infinite dimensional linear operators.

math.DS↗

arXiv2013

Spectra of stochastic adding machines based on Cantor Systems of numeration

In this paper, we define a stochastic adding machine based on Cantor Systems of numeration. We also compute the parts of spectra of the transition operator associated to this stochastic adding machine in different Banach spaces as $c_0, c$ and $l_α,\; 1 \leq α\leq +\infty$. These spectra are connected to fibered Julia sets.

math.PR↗

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

Spectrum of stochastic adding machines and fibered Julia sets

Consider the basic algorithm to perform the transformation n--> n+1 changing digits of the d-adic expansion of n one by one. We obtain a family of Markov chains on the non-negative integers through sucessive and independent applications of the algorithm modified by a parametrized stochastic rule that randomly prevents one of the steps in the algorithm to finish. The objects of study in this paper are the spectra of the transition operators of these Markov chains. The spectra of these Markov chains turn out to be fibered Julia sets of fibered polynomials. This enable us to analyze their topological and analytical properties with respect to the underlying parameters of the Markov chains.

math.DS↗

arXiv2008

Boundary of the Rauzy fractal sets in $\RR \times \CC$ generated by $P(x)=x^4-x^3-x^2-x-1$

We study the boundary of the 3-dimensional Rauzy fractal ${\mathcal E} \subset \RR \times \CC$ generated by the polynomial $P(x) = x^4-x^3-x^2-x-1$. The finite automaton characterizing the boundary of ${\mathcal E}$ is given explicitly. As a consequence we prove that the set ${\mathcal E}$ has 18 neighborhoods where 6 of them intersect the central tile ${\mathcal E}$ in a point. Our construction shows that the boundary is generated by an iterated function system starting with 2 compact sets.

math.NT↗