arXiv subjects

Bernat Espigule

Publications and source records attributed to Bernat Espigule.

6 recordsLinked to original sources

arXiv2026

Explicit interpolations among the Sierpi\'nski, Rauzy, and Apollonian gaskets

We study two explicit one-parameter families organized around the affine Sierpi\'nski gasket. The first is an affine-projective interpolation from the Sierpi\'nski gasket to the Rauzy gasket: the first-level hole is fixed throughout the family, the symbolic quotient remains the classical Sierpi\'nski 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 M\"obius deformation from the Sierpi\'nski 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.

math.DS↗

arXiv2026

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

We investigate the topology of connectedness loci, denoted as $M_n$, for a one-parameter family of collinear affine iterated function systems featuring equally spaced translations. These loci are arithmetically equivalent to the closures of roots of monic polynomials whose non-leading coefficients fall within a prescribed finite interval of integers. Our main theorem proves that for every integer $n \ge 2$, the connectedness locus $M_n$ contains infinitely many holes. While the $n=2$ case is equivalent to a known theorem by Calegari, Koch, and Walker, this paper establishes the proof for $n \ge 3$. To prove the existence of holes for larger alphabets, we construct a stationary family of finite-capture loops in the geometry of the associated difference attractor. Each loop surrounds a missing-center configuration, and a finite inverse-tree certificate rigorously demonstrates that the enclosed witness parameter lies outside the connectedness locus. Furthermore, we show that the sequence of witness parameters converges to a canonical algebraic boundary point -- termed the renormalization point, $\xi_n$ -- where infinitely many of these distinct holes accumulate. The paper's finite geometric checks are verified via exact algebraic certificates.

math.DS↗

arXiv2026

Finite capture and the closure of roots of restricted polynomials

We study how a countable algebraic root set passes to a fractal connectedness locus. Let $D_n=\{-n+1,-n+2,\ldots,n-1\}$, and let $R_n$ be the set of roots of monic polynomials whose non-leading coefficients lie in $D_n$. We study $\overline{R_n}\setminus\overline{\mathbb{D}}$. Outside the closed unit disk this set equals a connectedness locus $M_n$ for a collinear affine iterated function system, or equivalently the zero set of reciprocal power series $1+\sum_{k\ge1} d_k c^{-k}$ with $d_k\in D_n$. For non-real parameters in the lens $X_n=\{\,c\in\mathbb{C}\setminus\overline{\mathbb{D}}:\ |c\pm1|<\sqrt{2n}\,\}$ we construct a canonical trap and enclosure for the associated difference attractor and use them to define finite-capture sets $\Theta_k(n)$ for the marked point $2c$. Our main result is the uniform inclusion $\overline{\Theta_k(n)}\cap(X_n\setminus\mathbb{R})\subset\Theta_{k+2}(n)$ for every $k\ge0$. Consequently, $(M_n\cap X_n)\setminus\mathbb{R}$ is exactly the closure of the finite-capture locus. The paper combines explicit trap geometry with certified inverse search. Moreover, $M_n\setminus\mathbb{R}\subset X_n$ for every $n\ge20$, and this is sharp for $2\le n\le19$. Thus, for $n\ge20$, the non-real part of $\overline{R_n}\setminus\overline{\mathbb{D}}$ is exactly the closure of the finite-capture locus.

math.DS↗

arXiv2024

Collinear Fractals and Bandt's Conjecture

For a complex parameter $c$ outside the unit disk and an integer $n\ge2$, we examine the $n$-ary collinear fractal $E(c,n)$, defined as the attractor of the iterated function system $\{\mbox{$f_k \colon \mathbb{C} \longrightarrow \mathbb{C}$}\}_{k=1}^n$, where $f_k(z):=1+n-2k+c^{-1}z$. We investigate some topological features of the connectedness locus $\mathcal{M}_n$, similar to the Mandelbrot set, defined as the set of those $c$ for which $E(c,n)$ is connected. In particular, we provide a detailed answer to an open question posed by Calegari, Koch, and Walker in 2017. We also extend and refine the technique of the covering property by Solomyak and Xu to any $n\ge2$. We use it to show that a nontrivial portion of $\mathcal{M}_n$ is regular-closed. When $n\ge21$, we enhance this result by showing that, in fact, the whole $\mathcal{M}_n\setminus\mathbb{R}$ lies within the closure of its interior, thus proving that the generalized Bandt's conjecture is true.

math.DS↗

arXiv2019

Families of connected self-similar sets generated by complex trees

The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets. Systems of equations encoded by complex trees tip-to-tip equivalence relations are used to obtain one-parameter families of connected self-similar sets $F_\mathcal{A}(z)$. In order to study topological changes of $F_\mathcal{A}(z)$ in regions $\mathcal{R}\subset\mathbb{C}$ where these families are defined, we introduce a new kind of set $\mathcal{M}\subseteq\mathcal{R}$ which extends the usual notion of connectivity locus for a parameter space. Moreover we consider another set $\mathcal{M}_0\subseteq\mathcal{M}$ related to a special type of connectivity for which we provide a theorem. Among other things, the present theory provides a unified framework to families of self-similar sets traditionally studied as separate with elements $F_\mathcal{A}(z)$ disconnected for parameters $z\in\mathcal{R}\backslash\mathcal{M}$.

math.DS↗

arXiv2019

A family of fern-like ternary complex trees

A ternary complex tree related to the golden ratio is used to show how the theory of complex trees works. We use the topological set of this tree to obtain a parametric family of trees in one complex variable. Even though some real ferns and leaves are reminiscent to elements of our family of study, here we only consider the underlying mathematics. We provide aesthetically appealing examples and a map of the unstable set $\mathcal{M}$ for this family. Moreover we show that some elements found in the boundary of the unstable set $\mathcal{M}$ possess interesting algebraic properties, and we explain how to compute the Hausdorff dimension and the shortest path of self-similar sets described by trees found outside the interior of the unstable set $\mathcal{M}$.

math.DS↗