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 $2^m{q}^{-n}$ 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 $\lfloor n/2\rfloor <m<n$.

20 pages, 7 figures, 3 tables; updated to agree with published version