On trees invariant under edge contraction (Q502560)

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.