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Tree-like is not a transitive relation on paths

Jeremy Brazas, Gregory R. Conner, Paul Fabel, Curtis Kent

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.

Tree-like is not a transitive relation on paths

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.
Paper Structure (8 sections, 7 theorems, 6 equations, 5 figures)

This paper contains 8 sections, 7 theorems, 6 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Relation between types of tree-like
  3. Building dendrites and maps
  4. Parameterizing edges of a triangle
  5. The subdivision step
  6. The construction
  7. Limiting arguments
  8. Non-transitivity of tree-like equivalent

Key Result

Lemma 2.2

The equivalence relation $\sim_h$ induces an upper semi-continuous decomposition of $[0,1]$ such that the quotient $\pi: [0,1]\to E=[0,1]/\space\sim_h$ is a dendrite.

Figures (5)

  • Figure 1: Parameterizing $\triangle abc$ on $[r,s]$
  • Figure 2: Adding and subdividing edges
  • Figure 3: The maps
  • Figure 4: On the left are $E_2$ and $\tilde{E}_2$. On the right are $E_3$ and $\tilde{E}_3$.
  • Figure 5: On the left is a schematic for $g_2\circ \pi_2$ (in blue) and $\tilde{g}_2\circ \tilde{\pi}_2$ (in red). On the right is a schematic for $g_3\circ \pi_3$ (in blue) and $\tilde{g}_3\circ \tilde{\pi}_3$ (in red).

Theorems & Definitions (18)

  • Definition 1.1: HamblyLyons2010
  • Definition 1.4: BoedihardjoGengLyonsYang2016BrazasConnerFabelKent_preprint
  • Definition 2.1: Dendrites from height functions
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: Height functions from dendrites
  • proof
  • Remark 2.5
  • ...and 8 more