Substitution Rules for Higher-Dimensional Paperfolding Structures

TL;DR

This paper introduces a general substitution scheme for creating higher-dimensional paperfolding structures, proving their dynamical and spectral properties, and analyzing their complexity and topological invariants.

Contribution

It provides the first systematic method to generate and analyze higher-dimensional paperfolding structures with proven dynamical and spectral properties.

Findings

The substitution is primitive and has a coincidence, leading to pure point spectrum.

The dynamical system is strictly ergodic.

Topological invariants like Čech cohomology are computed for dimensions up to 2.

Abstract

We present a general scheme how to construct a substitution rule for generating $d$-dimensional analogues of the paperfolding structures. This substitution is proven to be primitive, so that the translation action on the hull forms a strictly ergodic dynamical system. The substitution admits a coincidence in the sense of Dekking, which implies that the dynamical system has pure point spectrum. The same then holds true also for the diffraction spectrum. The substitution also allows us to give estimates on the complexity of the paperfolding structures, and to determine topological invariants like the \v{C}ech cohomology groups of the hull for dimensions $d\le2$.