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A subperiodic tree whose intermediate branching number is strictly less than the lower intermediate growth rate

Pengfei Tang

Abstract

We construct an example of a subperiodic tree whose intermediate branching number is strictly less than the lower intermediate growth rate. This answers a question of Amir and Yang (2022) in the negative.

A subperiodic tree whose intermediate branching number is strictly less than the lower intermediate growth rate

Abstract

We construct an example of a subperiodic tree whose intermediate branching number is strictly less than the lower intermediate growth rate. This answers a question of Amir and Yang (2022) in the negative.
Paper Structure (8 sections, 3 theorems, 26 equations, 3 figures)

This paper contains 8 sections, 3 theorems, 26 equations, 3 figures.

Table of Contents

  1. Introduction and main result
  2. Various branching numbers and growth rates of infinite trees
  3. Subperiodic trees
  4. Proof of the main result
  5. Coding by trees
  6. The construction of our example
  7. The intermediate branching number and the intermediate growth rate of our example
  8. Concluding remarks

Key Result

Theorem 1.2

There exists a subperiodic tree $T$ with intermediate growth rate and

Figures (3)

  • Figure 1: The $1$-$3$ tree $T_{1,3}$ and its labeling.
  • Figure 2: The tree $T_{0}$ and its labeling.
  • Figure 3: A lexicographically minimal spanning tree of $\mathbb{Z}^2$.

Theorems & Definitions (9)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1: Coding by trees
  • Remark 2.2
  • Example 2.4
  • Proposition 2.6
  • proof
  • Proposition 3.1
  • proof