Combinatorics of the tautological lamination

Danny Calegari

Vol. 329 (2024), No. 1, 39–61

DOI: 10.2140/pjm.2024.329.39

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 $\lfloor n∕2\rfloor <m<n$.

Keywords

shift locus, elaminations, tautological lamination, bordered words, tau sequence

Mathematical Subject Classification

Primary: 37F10, 68R15

Milestones

Received: 4 November 2021

Revised: 27 March 2024

Accepted: 1 April 2024

Published: 12 June 2024