Substitutive Arnoux-Rauzy sequences have pure discrete spectrum

We prove that the symbolic dynamical system generated by a purely substitutive Arnoux-Rauzy sequence is measurably conjugate to a toral translation. The proof is based on an explicit construction of a fundamental domain with fractal boundary (a Rauzy fractal) for this toral translation.

💡 Research Summary

The paper addresses the long‑standing problem of determining the spectral type of symbolic dynamical systems generated by purely substitutive Arnoux‑Rauzy sequences. An Arnoux‑Rauzy sequence is defined via an S‑adic construction using three elementary substitutions σ₁, σ₂, σ₃ on a three‑letter alphabet. When a finite product σ of these substitutions contains each σ_i at least once, σ is a unimodular Pisot‑irreducible substitution. The authors prove that the associated symbolic dynamical system (X_σ, S) has pure discrete spectrum, i.e., it is measurably conjugate to a translation on the two‑dimensional torus T².

The proof proceeds through a geometric construction of a Rauzy fractal ℛ_σ, which serves as a fundamental domain for the toral translation. The key steps are:

1. Algebraic Setup: For a Pisot‑irreducible substitution σ, the incidence matrix M_σ has a dominant eigenvalue β (a Pisot number) with right eigenvector u_β and left eigenvector v_β. The contracting plane P_c orthogonal to v_β is defined, and the projection π_c : ℝ³ → P_c along u_β is used to embed symbolic data into Euclidean space.

2. Dual Substitutions: The authors introduce the dual substitution Σ = E₁*(σ), which acts on unit faces of the cubic lattice ℤ³ rather than on letters. Σ is completely determined by M_σ and the images of the three canonical faces. Crucially, Σ maps a discrete plane Γ_v to Γ_{M_σ v}. In particular, the discrete plane Γ_{v_β} associated with the contracting plane is invariant under Σ.

3. Construction of the Rauzy Fractal: Starting from the basic face set U =