Geometric representation of the infimax S-adic family

TL;DR

This paper develops a geometric framework for the infimax S-adic family of substitutions, extending Rauzy-Canterini-Siegel methods to analyze their minimal sets, fractals, and interval translation maps.

Contribution

It generalizes geometric realization techniques to the S-adic case, constructing Rauzy fractals and interval translation maps for the infimax family.

Findings

Constructed geometric realizations for the infimax S-adic family.

Proved the Rauzy fractal as the attractor of an interval translation map.

Established disjointness of subpieces in the Rauzy fractal.

Abstract

We construct geometric realizations for the infimax family of substitutions by generalizing the Rauzy-Canterini-Siegel method for a single substitution to the S-adic case. The composition of each countably infinite subcollection of substitutions from the family has an asymptotic fixed sequence whose shift orbit closure is an infimax minimal set $\Delta^+$. The subcollection of substitutions also generates an infinite Bratteli-Vershik diagram with prefix-suffix labeled edges. Paths in the diagram give the Dumont-Thomas expansion of sequences in $\Delta^+$ which in turn gives a projection onto the asymptotic stable direction of the infinite product of the Abelianization matrices. The projections of all sequences from $\Delta^+$ is the generalized Rauzy fractal which has subpieces corresponding to the images of symbolic cylinder sets. The intervals containing these subpieces are shown to be disjoint except at endpoints, and thus the induced map derived from the symbolic shift translates them. Therefore the process yields an Interval Translation Map, and the Rauzy fractal is proved to be its attractor.