A theorem of Roe and Strichartz on homogeneous trees

Let 𝔛 {\mathfrak{X}} be a homogeneous tree and let ℒ {\mathcal{L}} be the Laplace operator on 𝔛 {\mathfrak{X}} . In this paper, we address problems of the following form: Suppose that { f k } k ∈ ℤ {\{f_{k}\}_{k\in\mathbb{Z}}} is a doubly infinite sequence of functions in 𝔛 {\mathfrak{X}} such that for all k ∈ ℤ {k\in\mathbb{Z}} one has ℒ ⁢ f k = A ⁢ f k + 1 {\mathcal{L}f_{k}=Af_{k+1}} and ∥ f k ∥ ≤ M {\lVert f_{k}\rVert\leq M} for some constants A ∈ ℂ {A\in\mathbb{C}} , M > 0 {M>0} and a suitable norm ∥ ⋅ ∥ {\lVert\,\cdot\,\rVert} . From this hypothesis, we try to infer that f 0 {f_{0}} , and hence every f k {f_{k}} , is an eigenfunction of ℒ {\mathcal{L}} . Moreover, we express f 0 {f_{0}} as the Poisson transform of functions defined on the boundary of 𝔛 {\mathfrak{X}} .

Unitarization of the Horocyclic Radon Transform on Homogeneous Trees

Following previous work in the continuous setup, we construct the unitarization of the horocyclic Radon transform on a homogeneous tree X and we show that it intertwines the quasi regular representations of the group of isometries of X on the tree itself and on the space of horocycles.

An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

For harmonic functions v on the disc, it has been known for a long time that non-tangential boundedness a.e.is equivalent to finiteness a.e. of the integral of the area function of v (Lusin area theorem). This result also hold for functions that are non-tangentially bounded only in a measurable subset of the boundary, and has been extended to rank-one hyperbolic spaces, and also to infinite trees (homogeneous or not). No equivalent of the Lusin area theorem is known on higher rank symmetric spaces, with the exception of the degenerate higher rank case given by the cartesian product of rank-one hyperbolic spaces. Indeed, for products of two discs, an area theorem for jointly harmonic functions was proved by M.P. and P. Malliavin, who introduced a new area function; non-tangential boundedness a.e. is a sufficient condition, but not necessary, for the finiteness of this area integral. Their result was later extended to general products of rank-one hyperbolic spaces by Korányi and Putz. Here we prove an area theorem for jointly harmonic functions on the product of a finite number of infinite homogeneous trees; for the sake of simplicity, we give the proofs for the product of two trees. This could be the first step to an area theorem for Bruhat–Tits affine buildings, thereby shedding light on the higher rank continuous set-up.

Riesz means on homogeneous trees

Let 𝕋 be a homogeneous tree. We prove that if f ∈ Lp (𝕋), 1 ≤ p ≤ 2, then the Riesz means Sz R (f) converge to f everywhere as R → ∞, whenever Re z > 0.

On some threshold-one attractive interacting particle systems on homogeneous trees

We consider the threshold-one contact process, the threshold-one voter model and the threshold-one voter model with positive spontaneous death on homogeneous trees $\mathbb{T}_d$ , $d\ge 2$ . Mainly inspired by the corresponding arguments for the contact process, we prove that the complete convergence theorem holds for these three systems under strong survival. When the system survives weakly, complete convergence may also hold under certain transition and/or initial conditions.

Hausdorff and packing dimensions of Mandelbrot measure

We develop, in the context of the boundary of a supercritical Galton–Watson tree, a uniform version of large deviation estimate on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set [Formula: see text]. As an application, we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of a fractal set related to covering number on the Galton–Watson tree.