Related papers: Tree-like is not a transitive relation on paths
In [9] Kaimanovich introduced the concept of augmented tree on the symbolic space of a self-similar set. It is hyperbolic in the sense of Gromov, and it was shown in [13] that under the open set condition, a self-similar set can be…
We initiate the study of the forward and backward shifts on the Lipschitz space of a tree, $\mathcal L$, and on the little Lipshitz space of a tree, ${\mathcal L}_0$. We determine that the forward shift is bounded both on $\mathcal L$ and…
We first identify (up to linear isomorphism) the Lipschitz free spaces of quasiarcs. By decomposing quasiconformal trees into quasiarcs as done in an article of David, Eriksson-Bique, and Vellis, we then identify the Lipschitz free spaces…
A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. We study the geometry of these trees in two directions. First, we construct a…
Given a Lipschitz map $f$ from a cube into a metric space, we find several equivalent conditions for $f$ to have a Lipschitz factorization through a metric tree. As an application we prove a recent conjecture of David and Schul. The…
In \cite{LuLa13}, two of the authors initiated a study of Lipschitz equivalence of self-similar sets through the augmented trees, a class of hyperbolic graphs introduced by Kaimanovich \cite{Ka03} and developed by Lau and Wang…
In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. We extend…
We provide decidability and undecidability results on the model-checking problem for infinite tree structures. These tree structures are built from sequences of elements of infinite relational structures. More precisely, we deal with the…
We compute the Lipschitz-free spaces of subsets of the real line and characterize subsets of metric trees by the fact that their Lipschitz-free space is isometric to a subspace of $L_1$.
The signature of a path, introduced by K.T. Chen [5] in $1954$, has been extensively studied in recent years. The $2010$ paper [12] of Hambly and Lyons showed that the signature is injective on the space of continuous finite-variation paths…
Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar…
For a tree $T$, let $lp(T)$ be the number of different lengths of leaf to leaf paths in $T$. For a degree sequence $s$ of a tree, let ${\rm rad}(s)$ be the minimum radius of a tree with degree sequence $s$. Recently, Di Braccio,…
Assume that we embed the path $P_n$ as a subgraph of a $2$-dimensional grid, namely, $P_k \times P_l$. Given such an embedding, we consider the ordered set of subpaths $L_1, L_2, \ldots , L_m$ which are maximal straight segments in the…
We give a parity reversing involution on noncrossing trees that leads to a combinatorial interpretation of a formula on noncrossing trees and symmetric ternary trees in answer to a problem proposed by Hough. We use the representation of…
In 2005 Kawamura and Rambla, independently, constructed a metric counterexample to Wood's Conjecture from 1982. We exhibit a new nonmetric counterexample of a space $\hat L$, such that $C_0(\hat L,\mathbb{C})$ is almost transitive, and show…
We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a…
For a finitely generated group $G$, we introduce an asymmetric pseudometric on projectivized deformation spaces of $G$-trees, using stretching factors of $G$-equivariant Lipschitz maps, that generalizes the Lipschitz metric on Outer space…
On a transient weighted graph, there are two models of random walk which continue after reaching infinity: random interlacements, and random walk reflected off of infinity, recently introduced in arXiv:2506.18827 [math.PR]. We prove these…
We combine conditions found in [Wh] with results from [MPR] to show that quasi-isometries between uniformly discrete bounded geometry spaces that satisfy linear isoperimetric inequalities are within bounded distance to bilipschitz…
A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. In this paper we show that every quasiconformal tree bi-Lipschitz embeds in some…