Explicit interpolations among the Sierpiński, Rauzy, and Apollonian gaskets

Abstract

We study two explicit one-parameter families organized around the affine Sierpiński gasket. The first is an affine-projective interpolation from the Sierpiński gasket to the Rauzy gasket: the first-level hole is fixed throughout the family, the symbolic quotient remains the classical Sierpiński quotient, the associated three-dimensional stack is homeomorphic to $S\times[0,1]$, and the constant-address cells display a transition from uniform contraction to the non-uniform projective scaling of the Rauzy endpoint. The second is a Möbius deformation from the Sierpiński gasket to the equilateral Apollonian gasket: for every hyperbolic parameter the maps form a conformal iterated function system on a common disk, vary continuously on compact subintervals away from the parabolic endpoint, and admit an exact variational formula for the Hausdorff dimension. At the parabolic endpoint we give an explicit boundary-normalized Apollonian model whose distinguished side is the segment $[0,1]$; in this normalization two endpoint branches are $z/(z+1)$ and $1/(2-z)$, the same fractional transformations that occur on the distinguished Rauzy side. This yields a canonical Rauzy-Apollonian homeomorphism fixing the common side pointwise. The constructions provide an exact framework for comparing projective, affine, and conformal triangular fractals through explicit maps, finite-level approximants, and three-dimensional embeddings.