Self-similar real trees defined as fixed-points and their geometric properties

Abstract : We consider fixed-point equations for probability measures charging measured compact metric spaces that naturally yield continuum random trees. On the one hand, we study the existence, the uniqueness of the fixed-points and the convergence of the corresponding iterative schemes. On the other hand, we study the geometric properties of the random measured real trees that are fixed-points, in particular their fractal properties. We obtain bounds on the Minkowski and Hausdorff dimension, that are proved tight in a number of applications, including the very classical continuum random tree, but also for the dual trees of random recursive triangulations of the disk introduced by Curien and Le Gall [Ann Probab, vol. 39, 2011]. The method happens to be especially powerful to treat cases where the natural mass measure on the real tree only provides weak estimates on the Hausdorff dimension.
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Pré-publication, Document de travail
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Contributeur : Nicolas Broutin <>
Soumis le : mercredi 19 octobre 2016 - 16:26:46
Dernière modification le : jeudi 26 avril 2018 - 10:28:29

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  • HAL Id : hal-01384309, version 1
  • ARXIV : 1610.05331



Nicolas Broutin, Henning Sulzbach. Self-similar real trees defined as fixed-points and their geometric properties. 2016. 〈hal-01384309〉



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