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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https://hal.inria.fr/hal-01384309
Contributor : Nicolas Broutin <>
Submitted on : Wednesday, October 19, 2016 - 4:26:46 PM
Last modification on : Friday, November 16, 2018 - 5:33:00 PM

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

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

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