Finite capture and the closure of roots of restricted polynomials

Abstract

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 $Θ_k(n)$ for the marked point $2c$. Our main result is the uniform inclusion $\overline{Θ_k(n)}\cap(X_n\setminus\mathbb{R})\subsetΘ_{k+2}(n)$ 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.

Figures (12)

• Figure 1: Finite-capture filtration for $n=3$ ($N=5$). Top: connectedness locus $\mathcal{M}_n$. Middle: difference attractor $E(c,5)$ for $c=0.7+1.4i$ with $2c$ captured at depth $4$. Bottom: original attractor $E(c,3)$.
• Figure 2: Canonical coordinates straighten $\mathcal{P}_c(\mathcal{S},\mathcal{V})$ into a rectangle.
• Figure 3: Lens $\mathcal{X}_n=\{\,c\in\mathbb{C}\setminus\overline{\mathbb{D}}:\ |c\pm1|< \sqrt{2n}\,\}$, where the nearest admissible digit criterion is feasible (\ref{['prop:NE-feasible-lens']}).
• Figure 4: Canonical trap and first forward iterate in phase space for $n=3$ ($N=5$) and $c=0.7+1.4i$. The lighter region is $\mathcal{C}_0(c):=\mathcal{C}(c,N)$ and the darker region is $\mathcal{C}_1(c)=\mathcal{H}_c(\mathcal{C}_0(c))$. The inclusion $\mathcal{C}_0(c)\subset\mathcal{C}_1(c)$ illustrates the self-covering mechanism.
• Figure 5: First-quadrant view of the finite-capture set $\Theta_2(20)$. The darker region is $\Xi_2(20)\setminus\Theta_2(20)$; it contains $\partial\mathcal{M}_{20}\setminus\mathbb{R}$.

Theorems & Definitions (101)

• Theorem A: Two-step closure theorem
• Theorem B: Non-real closure in the lens
• Corollary B: Completion by the real trace in the lens
• Theorem C: Sharp threshold for the non-real locus
• Corollary C: Global description for n>=20
• Proposition 1
• proof
• Remark 1
• Remark 2: Basic symmetries
• Lemma 1