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Null frontier of the Tribonacci Rauzy fractal

Abstract

The frontier of the actual Tribonacci Rauzy fractal has planar Haar measure zero.

Theorem 1.1 (The original Rauzy frontier has volume zero).

Lean statement: D5/S3/Geometry/Rauzy/TribonacciNullBoundary.rauzyFractal_frontier_volume_zero

Proof. Machine-checked in Lean as D5/S3/Geometry/Rauzy/TribonacciNullBoundary.rauzyFractal_frontier_volume_zero (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every complex z belonging to the roots of the Tribonacci characteristic polynomial X^3 - X^2 - X - 1 and distinct from the real Tribonacci constant beta, volume(frontier(rauzyFractal z)) = 0. Here rauzyFractal z is the existing closure of finite exponent-zero admissible Boolean sums, with no three consecutive ones. Volume is the two-dimensional Haar measure on the complex plane. The three prefix branches are w mapped to z w, 1 + z^2 w, and 1 + z + z^3 w. The exact prefix cover, compactness and nonempty interior hold for this same set under these two root hypotheses. The root identities give z nonzero, norm(z) < 1 and beta norm(z)^2 = 1. The three branch area weights norm(z)^2, norm(z)^4 and norm(z)^6 are positive and sum to one by the reciprocal-power identity for beta. Multiplication by b scales planar Haar measure by norm(b)^2, from its real-linear determinant; translations preserve volume. All three branches are open and have common Lipschitz constant norm(z). The contracting-cover theorem constructs a whole interior cylinder and removes its positive weight from the frontier cover. Finite boundary mass then forces zero. Coverage, separation, tiling and a nullity premise are not additional hypotheses. The argument applies to both non-Perron roots.

References