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Regular closure of the Tribonacci Rauzy fractal

Abstract

The original Tribonacci Rauzy fractal is the closure of its interior under the two characteristic-root hypotheses.

Theorem 1.1 (Exact three-map cover).

Lean statement: D5/S3/Geometry/Rauzy/TribonacciRegularClosure.rauzyFractal_prefix_cover

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

Source. Repository-derived.

Commentary.

The existing name split equivalence reconstructs the three initial blocks through the coordinate identity for adjoining a Boolean digit. Lengths zero, one and two are handled directly by 0, 1, z and 1+z. The finite window is covered in reverse, and compact closed affine images extend the cover to the Rauzy closure. Thus every point in the fractal is the image of another point under one of the three prefix maps. Together with nonempty interior, this equality implies regular closure.

Theorem 1.2 (The actual Rauzy closure is regular closed).

Lean statement: D5/S3/Geometry/Rauzy/TribonacciRegularClosure.rauzyFractal_closure_interior

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

Source. Repository-derived.

Commentary.

Let z be a root of X^3 - X^2 - X - 1 distinct from the real Tribonacci constant beta. The set K is the closure of finite exponent-zero admissible Boolean sums. The split of admissible names into prefixes 0, 10 and 110 gives the reverse cover; the coordinate identity for adjoining a digit evaluates each prefix, and short names give 0, 1, z and 1+z. Compact images extend this to exact equality for the three prefix maps. The nonempty-interior theorem supplies an open patch. The secondary-root relation beta norm(z)^2 = 1 proves z is nonzero; the root norm bound gives one common contraction constant norm(z) for z, z^2 and z^3. Their affine maps are open by multiplication homeomorphisms and translations. The parametric compact metric theorem gives closure(interior K) = K. Exact self-coverage and nonempty interior both follow from the two stated conditions on z. The three maps are w ↦ z w, w ↦ 1 + z^2 w, and w ↦ 1 + z + z^3 w.

References