Combinatorics of the Tautological Lamination

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 $\lfloor n/2 \rfloor < m < n$.

Figures (7)

• Figure 1: Pinching a circle along a finite lamination to obtain a collection of smaller circles.
• Figure 2: A finite elamination approximating the Degree 3 Tautological Elamination for $z=2$, and the result of pinching
• Figure 3: Part of the degree 3 Shift locus (in blue) in a coordinate slice $f(z)=z^3+pz+1$.
• Figure 4: Tautological Laminations for $q=3,4,5,6$.
• Figure 5: The maximal component $K$ of $S^1 \mod \Lambda_T$

Theorems & Definitions (51)

• Proposition 2.1: Recursion
• Proposition 2.2: Closed form solution
• proof
• Definition 3.1: Ad hoc definition, degree 3
• Example 3.2: Depth 1
• Example 3.3: Depth 2
• Example 3.4: Depth 3
• Lemma 3.5: Division by $q$
• proof
• Definition 3.6