Fractal tiles induced by tent maps

The paper proves Theorem 1.1: for each special Pisot number α, the tent‑tile Fα is a compact set equal to the closure of its interior, by exhibiting an explicit correspondence with Rauzy fractals and invoking the standard result that Rauzy tiles are closures of their interiors (stated as Proposition 3.2 in the paper). The formal setup in the contracting representation space Kα (omitting the principal embedding) and the IFS fL,fR is consistent with the model’s setup. The paper’s Section 4 establishes, case by case, that each tent‑tile is either itself a Rauzy fractal (up to linear isomorphism) or is a union of invariant Rauzy pieces; this suffices to deduce Fα = cl(int Fα) from Proposition 3.2. See the definition of Fα via contractions in Kα and the main statement of Theorem 1.1, as well as the Rauzy–tent correspondence and Proposition 3.2 (fL,fR contractions and tent‑tile definition; Theorem 1.1; relations to Rauzy fractals; closure-of-interior for Rauzy tiles) . The model’s argument follows the same route: work in the contracting space, invoke the Rauzy correspondence, and transfer the closure-of-interior property by affine equivalence. Minor overstatements in the model’s write-up (claiming every tent‑tile is affinely equivalent to a single Rauzy fractal and that its translates always form a lattice tiling) go beyond the paper: the paper shows some tent‑tiles equal a single Rauzy fractal while others are unions of Rauzy pieces, and it reports two cases where neither of the two tiling types (tile alone vs. tile with reflection) was found. These side claims are not needed for the main conclusion Fα = cl(int Fα), which the paper establishes rigorously and the model also reaches via the same underlying idea (two tiling types and exceptions; overview of varieties) .