37 research outputs found
Dimension of the harmonic measure of non-homogeneous Cantor sets
No full text13 pages, 2 figuresWe prove that the dimension of the harmonic measure of the complementary of a translation-invariant type of Cantor sets as a continuous function of the parameters determining these sets. This results extend a previous one of the author and do not use ergodic theoretic tools, not applicable to our case
HAUSDORFF AND HARMONIC MEASURES ON NON-HOMOGENEOUS CANTOR SETS
No full textAbstract. We consider (not self-similar) Cantor sets defined by a sequence of piecewise linear functions. We prove that the dimension of the harmonic measure on such a set is strictly smaller than its Hausdorff dimension. Some Hausdorff measure estimates for these sets are also provided. 1. Introduction. Statement o
Hausdorff and harmonic measures on non-homogeneous Cantor sets
No full text25 pagesWe consider (not self-similar) Cantor sets defined by a sequence of piecewise linear functions. We prove that the dimension of the harmonic measure on such a set is strictly smaller than its Hausdorff dimension. Some Hausdorff measure estimates for these sets are also provided
Simulation of the growth of cities
No full textDans cette thèse nous proposons et nous mettons en application plusieurs modèles décrivant la croissance et la morphologie du tissu urbain. Le premier de ces modèles est issu de la percolation en gradient (correlée) déjà proposé de la littérature. Le second, inédit, fait appel à un équation différentielle stochastique. Nos modèles sont paramétrables : les paramètres que nous avons choisi d’appliquer sont naturels et tiennent compte de l’accessibilité des sites. Le résultat des simulations est conforme à la réalité du terrain. Par ailleurs, nous étudions la percolation en gradient: nous démontrons , suivant Nolin, que la frontière de cluster principal se situe dans un voisinage de la courbe critique et nous estimons ses longueurs et largeurs. Enfin, nous menons une étude du processus de croissance SLE. Nous calculons (preuve assistée par ordinateur) l’espérance des carrés des modules pour SLE2 and SLE6. Ces résultats sont liés à la conjecture de Bieberbach.In this thesis we propose and test models that describe the growth and morphology of cities. The first of these models is used from previously developed correlated gradient percolation model. The second model is related to a stochastic differential equation and has never been proposed before. Both models are parameterizable. The parameters we chose in applications are well justified by physical observations: proximily to axes and accessibility of sites. The result is consistent with actual data. We also study the gradient percolation as a mathematical object. We prove, following Nolin’s ideas, that the front of gradient percolation cluster is localised in a neighborhood of the critical curve with width and length depending on density gradient. Finally, we also study SLE growth processes. We calculate (computer assisted demonstration) the expected value of square of moduli for SLE2 and SLE6 related to the Bieberbach conjecture
On entropy and Hausdorff dimension of measures defined through a non-homogeneous Markov process
No full text13 pagesIn this work we study the Hausdorff dimension of measures whose weight distribution satisfies a markov non-homogeneous property. We prove, in particular, that the Hausdorff dimensions of this kind of measures coincide with their lower Rényi dimensions (entropy). Moreover, we show that the Tricot dimensions (packing dimension) equal the upper Rényi dimensions. As an application we get a continuity property of the Hausdorff dimension of the measures, when it is seen as a function of the distributed weights under the norm
On entropy and Hausdorff dimension of measures defined through a Markov process
No full textIn this work we study the Hausdorff dimension of measures whose weight distribution satisfies a markov non-homogeneous property. We prove, in particular, that the Hausdorff dimensions of this kind of measures coincide with their lower Rényi dimensions (entropy). Moreover, we show that the Tricot dimensions (packing dimension) equal the upper Rényi dimensions. As an application we get a continuity property of the Hausdorff dimension of the measures, when it is seen as a function of the distributed weights under the ` ∞ norm.
On relations between entropy and Hausdorff dimension of measures
No full textInternational audienceWe characterize probability measures whose Hausdorff dimension or packing dimension can be calculated by an entropy formula. In particular, we prove that such measures are unidimensional. We also construct examples of unidimensional measures for which entropy does not calculate the dimension
Dimensions of harmonic measures on non-autonomous Cantor sets
No full textWe consider Non Autonomous Conformal Iterative Function Systems (NACIFS)and their limit set. Our main concern is harmonic measure and its dimensions : Haus-dorff and Packing. We prove that this two dimensions are continuous under perturba-tions and that they verify Bowen’s and Manning’s type formulas. In order to do so weprove general results about measures, and more generally about positive functionals,defined on a symbolic space, developing tools from thermodynamical formalism in anon-autonomous setting
