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.Comment: 26pages, 2 figure

On the spectrum of stochastic perturbations of the shift and Julia sets

We extend the Killeen-Taylor study in \cite{KT} by investigating in different Banach spaces ($\ell^\alpha(\N), c_0(\N),c_c(\N)$) the point, continuous and residual spectra of stochastic perturbations of the shift operator associated to the stochastic adding machine in base 2 and in Fibonacci base. For the base 2, the spectra are connected to the Julia set of a quadratic map. In the Fibonacci case, the spectra involve the Julia set of an endomorphism of \C^2.Comment: 24 pages, 8 figures and 10 reference

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.Comment: 34 pages, 19 figure

A class of cubic Rauzy Fractals

In this paper, we study arithmetical and topological properties for a class of Rauzy fractals ${\mathcal R}_a$ given by the polynomial $x^3- ax^2+x-1$ where $a \geq 2$ is an integer. In particular, we prove the number of neighbors of ${\mathcal R}_a$ in the periodic tiling is equal to $8$. We also give explicitly an automaton that generates the boundary of ${\mathcal R}_a$. As a consequence, we prove that ${\mathcal R}_2$ is homeomorphic to a topological disk

« Le cercueil du vivant ». Un habitat algérien aux limites de l’insupportable

La dynamique de formation de nouvelles territorialités rurales, la manière dont est aujourd’hui habité le territoire rural, plus précisément la relation entre la morphologie de l’habitat et la dynamique du territoire dans lequel il s’inscrit, sont au centre de cet article. Dans la vallée du Saf-Saf (wilaya de Skikda), le territoire rural connaît aujourd’hui des problèmes de recomposition territoriale largement liés à l’état d’insécurité marquant la décennie 1990. Contraintes de quitter leurs terres et leurs demeures pour se rapprocher d’axes de développement plus sûrs, les populations rurales en ont été particulièrement touchées, dans leur mode de vie et d’habitat. À la diversité harmonieuse du territoire rural algérien s’est substituée une diversité confuse de territoires qu’interpénètrent les rejaillissements de la crise urbaine sur les campagnes et les répercutions humaines et spatiales des troubles de la période dite de tragédie nationale.The dynamic formation of new rural territori-alities and the way in which rural territorialities are inhabited today, more precisely the relationship between the morphology of the habitat and the dynamics of the territory in which it is located, are the focal issues of this article. In the valley of Saf-Saf (wilaya of Skikda), rural territories are currently experiencing problems of territorial recomposition largely related to the state of insecurity marking the decade of the 1990s. Forced to leave their home ground and residences to approach more secure axes of development, the rural populations were particularly victimized by this insecurity, both as regards their habitat and their way of life. The harmonious diversity of rural Algeria has been replaced by a confused diversity of territories whose reflections of the urban crisis interpenetrate in the countryside and reflect the human and spatial disorders of the “national tragedy.