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.

• The paper constructs two explicit one-parameter iterated function systems connecting the Sierpiński gasket to the Rauzy and Apollonian gaskets through affine-projective and Möbius deformations.
• It proves that the systems share the Sierpiński symbolic quotient, producing canonical homeomorphisms, a continuous three-dimensional stack for the Rauzy family, and Hausdorff continuity across the affine-projective interpolation.
• It derives exact Hausdorff-dimension formulas for affine-projective fibers and records the Apollonian endpoint dimension as approximately 1.305686728049877, while leaving parameter regularity and parabolic endpoint continuity open.

Overview

This paper constructs two explicit one-parameter families of iterated function systems (IFSs) anchored at the affine Sierpiński gasket, each reaching a distinct classical triangular fractal at an endpoint. The first family is an affine-projective interpolation on the standard simplex whose endpoint at $t=1$ is the Rauzy gasket; the second is a Möbius deformation on the unit disk whose parabolic endpoint at $t=1$ is the equilateral Apollonian gasket. The paper's contribution is to keep the comparison between these three geometries entirely at the level of explicit maps, symbolic codings, finite-level approximants, and three-dimensional embeddings, rather than relying on abstract existence results. A central structural finding is that both endpoint systems share the same symbolic quotient as the Sierpiński gasket — the quotient of $\Sigma=\{1,2,3\}^{\mathbb N}$ by the vertex identifications $wij^\infty\sim wji^\infty$ — which yields canonical homeomorphisms throughout both families.

The Rauzy–Sierpiński affine-projective family

The interpolating maps are defined via matrices $M_i(t)$ acting projectively on the simplex $T$, with chart formulas such as $f_{1,t}(u,v)=(u,v)/(2-t+t(u+v))$. At $t=0$ these recover the three homotheties of the Sierpiński gasket; at $t=1$ they are the classical Rauzy inverse branches of Arnoux–Starosta (2606.23278). Two features hold for all $t\in[0,1]$: the first-level decomposition $t=1$0 is parameter-independent, so the first-level hole never moves; and the pairwise intersections of first-level cells are exactly the midpoints $t=1$1, which forces the address relation to be constant along the family.

For every $t=1$2 each branch is $t=1$3-Lipschitz in the sup norm, so Hutchinson's theorem supplies a unique attractor $t=1$4. The coding map $t=1$5 exists, and the paper proves that $t=1$6 holds if and only if $t=1$7 — the same equivalence for every $t=1$8, with the Rauzy endpoint coded through the same quotient. Consequently there are canonical homeomorphisms $t=1$9 matching symbolic addresses. The continuity argument extends to the full interval: finite cylinders vary continuously in $\Sigma=\{1,2,3\}^{\mathbb N}$0, and cylinder diameters collapse to zero at the Rauzy endpoint, giving joint continuity of $\Sigma=\{1,2,3\}^{\mathbb N}$1 on $\Sigma=\{1,2,3\}^{\mathbb N}$2 and Hausdorff continuity of $\Sigma=\{1,2,3\}^{\mathbb N}$3 on all of $\Sigma=\{1,2,3\}^{\mathbb N}$4.

The most transparent signature of the degeneration appears along constant-address rays. For the branch $\Sigma=\{1,2,3\}^{\mathbb N}$5, the closed formula

$\Sigma=\{1,2,3\}^{\mathbb N}$6

governs the size of the $\Sigma=\{1,2,3\}^{\mathbb N}$7-th iterate image $\Sigma=\{1,2,3\}^{\mathbb N}$8. Thus repeated-cell collapse is exponential at rate $\Sigma=\{1,2,3\}^{\mathbb N}$9 for every contracting fiber but polynomial $wij^\infty\sim wji^\infty$0 at the Rauzy endpoint — an exact dichotomy reflecting the passage from uniform contraction to non-uniform projective scaling.

Assembling the fibers into $wij^\infty\sim wji^\infty$1 at heights $wij^\infty\sim wji^\infty$2, the resulting compact set $wij^\infty\sim wji^\infty$3 is homeomorphic to $wij^\infty\sim wji^\infty$4; injectivity across levels follows from the height coordinate, and within each level from the constancy of the coding quotient. This answers, in an explicit form, the problem posed by Arnoux at ICERM 2019 of relating the Rauzy gasket to the Sierpiński geometry by a three-dimensional model.

Exact Hausdorff dimension of the intermediate fibers

For $wij^\infty\sim wji^\infty$5, after normalizing $wij^\infty\sim wji^\infty$6 with $wij^\infty\sim wji^\infty$7, the fiber dimension equals the projective affinity dimension $wij^\infty\sim wji^\infty$8 of the semigroup $wij^\infty\sim wji^\infty$9. The proof reduces the intermediate case to Jurga's dimension theorem for positive $M_i(t)$0-semigroups satisfying the strong open set condition (2606.23278), via three verifications:

• Positive conjugation: an explicit matrix $M_i(t)$1 conjugates all generators into $M_i(t)$2 for $M_i(t)$3 close enough.
• Zariski density: strong irreducibility (via limiting rank-one matrices with independent rows) plus absence of invariant quadratic forms rules out the $M_i(t)$4 symmetric-square alternative, forcing Zariski closure equal to $M_i(t)$5.
• No ambient truncation: a direct singular-value estimate gives $M_i(t)$6, uniformly in $M_i(t)$7, so submultiplicativity implies $M_i(t)$8 always.

The resulting dimension profile is:

$M_i(t)$9

with the endpoint value at $T$0 supplied by Jurga's theorem for the Rauzy gasket. The formula is exact but implicit: regularity, monotonicity, and effective computation of $T$1 remain open.

The Sierpiński–Apollonian Möbius deformation

The conformal family acts on the closed unit disk with fixed vertices $T$2, $T$3, via

$T$4

where $T$5 and $T$6. At $T$7 these are the similarities $T$8; at $T$9 they are parabolic with neutral fixed points at the vertices, generating the equilateral Apollonian gasket. Each $f_{1,t}(u,v)=(u,v)/(2-t+t(u+v))$0 preserves the disk and is Lipschitz with explicit constant $f_{1,t}(u,v)=(u,v)/(2-t+t(u+v))$1 for $f_{1,t}(u,v)=(u,v)/(2-t+t(u+v))$2, with $f_{1,t}(u,v)=(u,v)/(2-t+t(u+v))$3 as $f_{1,t}(u,v)=(u,v)/(2-t+t(u+v))$4. Hence unique compact attractors $f_{1,t}(u,v)=(u,v)/(2-t+t(u+v))$5 exist for $f_{1,t}(u,v)=(u,v)/(2-t+t(u+v))$6, and the attractor map is continuous on every truncated interval $f_{1,t}(u,v)=(u,v)/(2-t+t(u+v))$7.

Because each branch extends holomorphically past the disk (poles lie strictly outside for $f_{1,t}(u,v)=(u,v)/(2-t+t(u+v))$8), the family is a conformal IFS in the sense of Feng–Hu, yielding for every ergodic measure the exact-dimensional formula $f_{1,t}(u,v)=(u,v)/(2-t+t(u+v))$9 and the variational principle

$t=0$0

At the parabolic endpoint the paper quotes the Vytnova–Wormell computation:

$t=0$1

the sharpest numerical determination of the Apollonian dimension currently available (2606.23278).

Boundary normalization and the Rauzy–Apollonian homeomorphism

A specific Möbius map $t=0$2 sends the equilateral endpoint to an upper-half-plane model $t=0$3 whose distinguished side is $t=0$4 and whose adjacent outer circles have radius $t=0$5 centered at $t=0$6 and $t=0$7. In this normalization two endpoint branches restrict to the segment as

$t=0$8

which are precisely the restrictions of the corresponding Rauzy side branches $t=0$9 and $t=1$0. Matching the common Sierpiński address quotient then produces a canonical homeomorphism $t=1$1 that fixes the distinguished side pointwise. This separates exact topological equivalence from the genuinely different projective versus conformal dynamics of the two endpoints: the endpoints are not dynamically identical, but their distinguished sides carry identical one-dimensional codings.

Limitations and open questions

Several qualifications are stated explicitly in the paper. The dimension formula for the affine-projective fibers is implicit through $t=1$2, and its analytic behavior in $t=1$3 is unaddressed. On the conformal side, continuity of $t=1$4 is proved only away from the parabolic endpoint; finite-level computations suggest $t=1$5 in the Hausdorff metric as $t=1$6, but this endpoint continuity is not proved and is not used in any result. The variational dimension formula for hyperbolic fibers is not evaluated in closed form as a function of $t=1$7, in contrast to the much subtler parabolic computation at $t=1$8. Finally, the supplementary side-fixed visualization family $t=1$9 carries no formal dynamical assertion beyond its role as an illustration.

Conclusion

The paper provides an exact, fully explicit framework connecting three triangular fractals — projective (Rauzy), affine (Sierpiński), and conformal (Apollonian) — through shared symbolic quotients, parameter-independent first-level combinatorics, a product stack $t\in[0,1]$0, exact dimension formulas on all hyperbolic fibers, and a side-fixing homeomorphism between the Rauzy and boundary-normalized Apollonian endpoints. The remaining gaps — effective dependence of the affinity dimension on $t\in[0,1]$1 and Hausdorff continuity at the parabolic Apollonian endpoint — are precisely identified questions left open by the constructions.