Combinatorics of the Tautological Lamination

TL;DR

This paper studies the combinatorial structure of the Tautological Lamination in holomorphic dynamics, providing recursive formulas and gap theorems for its partition counts, revealing intricate patterns in its geometric model.

Contribution

It introduces a recursion formula for partition counts and establishes a gap theorem, advancing understanding of the lamination's combinatorial properties.

Findings

Derived a recursion formula for $N_q(n,0)$

Proved the gap theorem: $N_q(n,n)=1$ and $N_q(n,m)=0$ for certain m

Uncovered structural patterns in the lamination's partitions

Abstract

The Tautological Lamination arises in holomorphic dynamics as a combinatorial model for the geometry of 1-dimensional slices of the Shift Locus. In each degree $q$ the tautological lamination defines an iterated sequence of partitions of $1$ (one for each integer $n$) into numbers of the form $2^m q^{-n}$. Denote by $N_q(n,m)$ the number of times $2^mq^{-n}$ arises in the $n$th partition. We prove a recursion formula for $N_q(n,0)$, and a gap theorem: $N_q(n,n)=1$ and $N_q(n,m)=0$ for $\lfloor n/2 \rfloor < m < n$.