Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On continuum real trees of circle maps and their graphs

Maik Gröger, Sascha Troscheit

TL;DR

This paper establishes a precise link between the fractal geometry of real trees arising from circle maps and the analytic variation of their contour functions. By providing a concise proof of Picard's theorem, it shows that the upper box dimension of the tree $\mathcal{T}(f)$ equals the variation index $I(f)$, connecting dimension to Hölder-type regularity via packings and discretised variations. It then introduces upper/lower variation contents and indices, proving a variational principle that time-changes can maximize (or bound) the graph dimension of the contour, with sharp equality $\overline{\dim}_{\mathrm{B}}\Gamma(f\circ\tau)=2-1/I(f)$ for some time-change $\tau$. The work also discusses modified dimensions to achieve lower bounds and provides examples demonstrating sharpness, illustrating how analytic regularity controls the geometry of both trees and their graphs in the circle-mapping setting.

Abstract

The Brownian continuum tree was extensively studied in the 90s as a universal random metric space. One construction obtains the continuum tree by a change of metric from an excursion function (or continuous circle mapping) on $[0,1]$. This change of metric can be applied to all excursion functions, and generally to continuous circle mappings. In 2008, Picard proved that the dimension theory of the tree is connected to its associated contour function: the upper box dimension of the continuum tree coincides with the variation index of the contour function. In this article we give a short and direct proof of Picard's theorem through the study of packings. We develop related and equivalent notions of variations and variation indices and study their basic properties. Finally, we link the dimension theory of the tree with the dimension theory of the graph of its contour function.

On continuum real trees of circle maps and their graphs

TL;DR

This paper establishes a precise link between the fractal geometry of real trees arising from circle maps and the analytic variation of their contour functions. By providing a concise proof of Picard's theorem, it shows that the upper box dimension of the tree equals the variation index , connecting dimension to Hölder-type regularity via packings and discretised variations. It then introduces upper/lower variation contents and indices, proving a variational principle that time-changes can maximize (or bound) the graph dimension of the contour, with sharp equality for some time-change . The work also discusses modified dimensions to achieve lower bounds and provides examples demonstrating sharpness, illustrating how analytic regularity controls the geometry of both trees and their graphs in the circle-mapping setting.

Abstract

The Brownian continuum tree was extensively studied in the 90s as a universal random metric space. One construction obtains the continuum tree by a change of metric from an excursion function (or continuous circle mapping) on . This change of metric can be applied to all excursion functions, and generally to continuous circle mappings. In 2008, Picard proved that the dimension theory of the tree is connected to its associated contour function: the upper box dimension of the continuum tree coincides with the variation index of the contour function. In this article we give a short and direct proof of Picard's theorem through the study of packings. We develop related and equivalent notions of variations and variation indices and study their basic properties. Finally, we link the dimension theory of the tree with the dimension theory of the graph of its contour function.
Paper Structure (19 sections, 14 theorems, 82 equations, 2 figures)

This paper contains 19 sections, 14 theorems, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organisation.
  3. Acknowledgements.
  4. Definitions: Real trees of circle mappings
  5. Remark.
  6. Bounded variation and p-variation
  7. Dimension theory
  8. The box dimension
  9. The modified box dimension
  10. Picard's theorem: Relating box dimension and p-variation
  11. Discretised variations and variational contents
  12. Remark.
  13. Dimensions of Graphs and Trees
  14. Organisation.
  15. A maximising time change
  16. ...and 4 more sections

Key Result

Proposition 2.6

Let $f\in C(\mathop{\mathrm{\mathbb{S}}}\nolimits^1)$. The following two statements hold:

Figures (2)

  • Figure 1: The tree space associated with a very smooth function. The identified regions in the tree are captured by colour coding.
  • Figure 2: A Brownian excursion realizing the Brownian continuum random tree.

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Proposition 2.8
  • proof
  • ...and 22 more