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On trees invariant under edge contraction. (Au sujet des arbres invariants par contraction de leurs arêtes.) (English. French summary) Zbl 1364.60105
Consider a random rooted, locally finite tree $$T=(V,E,\rho)$$ with finitely many ends, and denote by $$\text{Spine}(T)$$ the union of ends of $$T$$ or $$\{\rho\}$$, if $$T$$ has no end. Take $$p,q\in(0,1)$$, set $$V_0= V\setminus\mathrm{Spine}(T)$$, $$V_1= \mathrm{Spine}(T)\setminus\rho$$, and define the contracted tree $$C_{p,q}(T,V')$$ as the tree with root $$\rho$$ and vertex set $$V'\subset V$$ containing $$\rho$$, every vertex in $$V_0$$ independently with probability $$p$$, and every vertex in $$V_1$$ independently with probability $$q$$. The partial order on $$V'$$ is the restriction of the partial order on $$V$$. The tree $$T$$ is called $$(p,q)$$-self-similar, if it equals $$C_{p,q}(T,V')$$ in distribution.
Next, consider the space of complete, locally compact, rooted, measured $$\mathbb{R}$$-trees $${\mathcal T}$$-$$({\mathcal V},d,\rho,\mu)$$, with $$\mu$$ bounded, modulo equivalence with respect to root- and measure-preserving isometries. Endow this space with the Gromov-Hausdorff-Prokhorov topology and the Borel $$\sigma$$-field induced by it. Restrict the space to trees with finitely many open ends and $$\mu$$ dominating the length measure $$\ell_{{\mathcal T}}$$. Define the spine of $${\mathcal T}$$ as the subset of vertices which lie on an end, and the rescaled tree $${\mathcal S}_{p,q}({\mathcal T})$$ as obtained from $${\mathcal T}$$ by shrinking distances on the spine by the factor $$p$$ and off the spine by a factor $$q$$, and scaling the component $$\mu$$-$$\ell_{{\mathcal T}}$$ of $$\mu$$ by the factor $$p$$. The tree $${\mathcal T}$$ is called $$(p,q)$$-self-similar, if it is equal in distribution to $${\mathcal S}_{p,q}({\mathcal T})$$.
A one-to-one correspondence of $$(p,q)$$-self-similar discrete trees and $$(p,q)$$-self-similar $$\mathbb{R}$$-trees is obtained via the following discretization of the latter: let $$V^{(0)}$$ be the set of atoms of a Poisson process on $${\mathcal V}$$ with intensity measure $$\ell_{{\mathcal T}}$$ and $$V^{(1)}$$ the multiset of atoms of a Poisson process on $${\mathcal V}$$ with intensity measure $$\mu$$-$$\ell_{{\mathcal T}}$$. Define $$T$$ as the rooted tree with vertex set $$V^{(0)}\cup V^{(1)}$$ and $$v$$ ancestor of $$w$$ in $$T$$ if and only if it is an ancestor of $$w$$ in $${\mathcal T}$$ and $$v\in V^{(0)}$$. As an example, the class of trees which are invariant with respect to translation along the spine is discussed. In the case of self-similar trees consisting of a single spine to which i.i.d. subtrees are attached, the construction of the corresponding $$\mathbb{R}$$-trees is related to the quasi-stationary distributions of linear-fractional subcritical Bienaymé-Galton-Watson processes.
##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60G18 Self-similar stochastic processes 60B10 Convergence of probability measures
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