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Infinitely many holes in connectedness loci for collinear affine iterated function systems

This paper proves that for every integer $n \ge 3$ , the connectedness locus of a specific family of collinear affine iterated function systems contains infinitely many holes that accumulate at a canonical renormalization point, thereby extending a known result for the $n=2$ case through the construction of finite-capture loops and algebraic certificates.

Original authors: Bernat Espigule

Published 2026-06-02

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CC BY 4.0

Original authors: Bernat Espigule

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect designing a building made of fractals. You have a set of rules (mathematical formulas) that tell you how to shrink and shift pieces of the building to create a complex, self-repeating shape called an "attractor."

Usually, if you tweak the rules slightly, the building stays in one piece. But sometimes, if you change a specific number (a parameter), the building suddenly splits into two separate islands, or even shatters into a million tiny dust motes. The "Connectedness Locus" is essentially a map of all the rule-settings where the building stays in one single, connected piece.

This paper is about a specific type of fractal building where the pieces are lined up in a straight row (collinear). The author, Bernat Espigule, asks a simple question: Does the map of "connected settings" look like a solid blob, or is it full of holes?

Here is the story of what he found, explained without the heavy math:

1. The Map of "Connectedness"

Think of the "Connectedness Locus" as a safety zone on a map. If you pick a number inside this zone, your fractal building is safe and connected. If you pick a number outside, it falls apart.

For a long time, mathematicians knew that for the simplest version of this problem (using just two rules), this safety zone wasn't a solid blob. It had holes. Imagine a Swiss cheese where the holes are actually tiny islands of "broken" buildings surrounded by "connected" ones.

The big question was: Does this happen when you use more than two rules (three, four, or more)?

2. The Discovery: Infinite Holes

The paper proves that yes, there are infinitely many holes, no matter how many rules you use (as long as there are at least two).

It's not just one or two holes; it's an endless pattern of them, getting smaller and smaller, clustering together like a swarm of bees.

3. How They Found the Holes: The "Ghost" Strategy

To prove these holes exist, the author uses a clever trick involving a "Ghost Parallelogram."

The Setup: Imagine you have four specific blueprints (called "charts") that tell you where the building is connected. These four blueprints are arranged in a square.
The Center: Usually, if you look at the exact center of this square, there should be a fifth blueprint that fills the gap, making the whole area solid.
The Ghost: In this specific mathematical setup, the "center blueprint" is a ghost. It looks like it should be there, but it's actually missing because the rules don't allow it.
The Result: Because the center is missing, the four surrounding blueprints form a ring (a loop) of "connected" settings. Inside that ring, there is a tiny island of "broken" settings. This island is a hole.

4. The "Stationary Transport" Machine

The author didn't just find one hole; he found a machine to make infinite holes.

He discovered a mathematical "transport operator." Think of it like a copy machine that takes the first hole he found and creates a slightly larger, slightly more complex version of it. Then, it takes that version and makes another.

He can keep pressing "copy" forever.
Each time he presses it, he generates a new, distinct hole in the map.
This proves there isn't just a finite number of holes, but an infinite number.

5. The Destination: The "Renormalization Point"

As the author keeps pressing "copy" to make more and more holes, they don't just scatter randomly. They all start marching toward a single, specific destination point on the map.

This point is a special number (an algebraic number) that acts like a magnet.
The holes get smaller and smaller as they get closer to this point.
Eventually, an infinite number of holes pile up right at this single spot. The author calls this the "renormalization point."

Summary

In simple terms, the paper shows that for these straight-line fractal systems, the "safety zone" where the shape stays connected is not a solid island. It is a porous sponge with infinitely many holes inside it.

The author built a mathematical machine that generates these holes one by one, proving they never stop appearing, and showing that they all eventually crowd together at one specific, mathematically precise location. This confirms a long-standing guess that these complex fractal maps are much more "holey" and intricate than previously thought.

Technical Summary: Infinitely Many Holes in Connectedness Loci for Collinear Affine Iterated Function Systems

Problem Statement
This paper investigates the topology of connectedness loci for a specific family of affine iterated function systems (IFS). The systems are defined by $n$ collinear maps on the complex plane with equally spaced translations. Specifically, for an integer $n \ge 2$ , the IFS is given by $f_t(z) = t + z/c$ where $t$ ranges over the set $A_n = \{-n+1, -n+3, \dots, n-1\}$ . The connectedness locus $M_n$ is the set of parameters $c \in \mathbb{C} \setminus \mathbb{D}$ (where $\mathbb{D}$ is the open unit disk) for which the attractor $E(c, n)$ is connected.

While the case $n=2$ (two maps) was previously studied by Calegari, Koch, and Walker, who proved the existence of infinitely many "holes" (bounded connected components of the complement of the locus) accumulating at a renormalization point, the status of $M_n$ for $n \ge 3$ was an open conjecture. This paper aims to prove that for every $n \ge 2$ , the locus $M_n$ contains infinitely many holes.

Methodology and Framework
The authors employ a combination of arithmetic root theory and geometric "finite-capture" techniques derived from a companion preprint [4].

Arithmetic Formulation: The connectedness locus $M_n$ is characterized as the closure of the roots of monic polynomials whose non-leading coefficients lie in the integer interval $D_n = \{-n+1, \dots, n-1\}$ . This equivalence relies on a "marked-point" criterion: $c \in M_n$ if and only if $2c$ belongs to the difference attractor $E(c, N)$ , where $N = 2n-1$ .
Geometric Trap and Enclosure: The proof utilizes a canonical "trap" $C_0(c)$ $C_{0} (c)$ (an open parallelogram centered at 0) and a "closed enclosure" $E_{enc}(c, N)$ $E_{e n c} (c, N)$ .
- Interior Certification: If a parameter $c$ allows a finite inverse orbit of $2c$ to land inside the trap $C_0(c)$ , then $c$ lies in the interior of $M_n$ .
- Exterior Certification: If a finite inverse tree of $2c$ becomes empty (i.e., no admissible inverse orbit remains within the enclosure), then $c \notin M_n$ .
Loop Witness Construction: To prove the existence of a hole, the authors construct a "loop witness" $(\Gamma, c^*)$ . This consists of a Jordan curve $\Gamma \subset M_n$ and a witness parameter $c^*$ inside the bounded component of $\Gamma$ such that $c^* \notin M_n$ . The existence of such a pair implies a hole exists within $\Gamma$ .

Key Contributions and Construction
The core of the paper is the construction of a "stationary family" of holes for $n \ge 3$ .

Ghost Parallelograms: The authors define a "ghost parallelogram" of depth $m$ as a quintuple of words $(d^{(1)}, d^{(2)}, d^{(3)}, d^{(4)}; w)$ where the four corner words are admissible (belong to the alphabet $A_N$ ), but the midpoint word $w$ is inadmissible (contains an odd digit, thus not in $A_N$ ).
Base Configuration: For each $n \ge 3$ , a specific set of four admissible words of length 18 is constructed. Their midpoint word $w_1$ has an odd digit at the 10th position, making it a ghost word. The boundaries of the corresponding four charts form a Jordan curve $\Gamma_{n,1}$ in $M_n$ .
Stationary Transport: A renormalization operator $R$ is defined that maps words of length $m$ to length $m+3$ . This operator preserves the midpoint relation and the "ghost" property (the first odd digit shifts but remains odd). By iterating this operator, the authors generate an infinite sequence of ghost parallelograms of increasing depth $m_k = 3k + 15$ .
Witness Parameters: For each level $k$ $k$ , the midpoint word $w_k$ $w_{k}$ defines a polynomial $W_{w_k}(c)$ $W_{w_{k}} (c)$ . The authors prove that within a specific "witness box" $B_n$ $B_{n}$ , there exists a unique root $c_{n,k}$ $c_{n, k}$ of $W_{w_k}$ $W_{w_{k}}$ .
- Using Rouché's theorem and explicit algebraic bounds, they show that $c_{n,k}$ lies inside the Jordan curve $\Gamma_{n,k}$ formed by the boundaries of the four admissible charts at level $k$ .
- They further prove that $c_{n,k}$ is not in $M_n$ because its inverse tree becomes empty at a finite depth (specifically, $T_{m_k-1}(c_{n,k}) = \emptyset$ ).

Results

Infinitely Many Holes: For every $n \ge 2$ $n \geq 2$ , the connectedness locus $M_n$ $M_{n}$ has infinitely many holes.
- For $n=2$ , the result follows from the inversion of the parameter and the known work of Calegari–Koch–Walker.
- For $n \ge 3$ , the result is proven by constructing the infinite sequence of disjoint holes $(\Gamma_{n,k}, c_{n,k})$ for odd $k$ .
Algebraic Accumulation Point: The sequence of witness parameters $c_{n,k}$ $c_{n, k}$ converges to a unique limit point $\xi_n$ $ξ_{n}$ as $k \to \infty$ $k \to \infty$ .
- This limit $\xi_n$ is the unique zero of an explicit degree-13 polynomial $Q_n(c)$ within the witness box $B_n$ .
- $\xi_n$ lies on the boundary of $M_n$ ( $\xi_n \in \partial M_n$ ) and is an algebraic number of degree at most 13.
- Every neighborhood of $\xi_n$ contains infinitely many distinct holes of $M_n$ .

Significance and Claims
The paper claims to resolve a conjecture regarding the topology of connectedness loci for collinear affine IFS with more than two maps.

Generalization: It extends the known phenomenon of "infinitely many holes" from the two-map case ( $n=2$ ) to all $n \ge 3$ .
Mechanism: Unlike the $n=2$ case where the mechanism is tied to specific two-map geometry, the paper demonstrates a "stationary" mechanism based on the transport of a "ghost vacancy" (a missing center chart due to an inadmissible midpoint word). This mechanism is uniform across all $n \ge 3$ .
Certification: The proofs rely on exact algebraic certificates (sign checks, root counts, and non-intersection verifications) rather than visual inspection, ensuring the holes are rigorously established.
Limit Point: The identification of a canonical algebraic boundary point $\xi_n$ where these holes accumulate provides a precise structural description of the renormalization behavior for these systems.

The paper does not propose new applications or experimental setups; its contribution is strictly theoretical, advancing the understanding of the fractal topology of parameter spaces in iterated function systems.