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

Abstract

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.