Tree-like is not a transitive relation on paths

TL;DR

This paper demonstrates that the property of being tree-like, when extended beyond Lipschitz paths, does not form an equivalence relation due to the loss of transitivity, as shown through a fractal counterexample.

Contribution

It proves that the non-Lipschitz tree-like relation is not transitive, contrasting with the Lipschitz case, and provides an explicit fractal example.

Findings

Non-Lipschitz tree-like relation is not transitive

Counterexample based on a fractal construction

Lipschitz condition is crucial for transitivity

Abstract

The notions of tree-like loop and Lipschitz tree-like loop were introduced by Hambly and Lyons in their 2010 Annals of Mathematics paper. They showed that the Lipschitz tree-like property determines an equivalence relation on the set of paths of bounded variation in a given metric space and then asked if this notion could be extended to paths without the Lipschitz requirement. We show that after eliminating the Lipschitz requirement, the resulting relation is no longer transitive and thus is not an equivalence relation. The counterexample is obtained by analyzing an explicit fractal construction in the plane.