Contact multiplicity and a sharp six-map bound for collinear self-similar dendrites

Abstract

Let $K\subset\mathbb{C}$ be the attractor of $n\ge2$ similarities $z\mapstoλz+b_k$, $0<|λ|<1$, whose fixed points are evenly spaced on a line, and suppose that $K$ does not lie in a line. We prove that $K$ can be a dendrite only if $n\le6$, and that such dendrites exist for every $2\le n\le6$; no separation or finite-intersection hypothesis is imposed. Hence, for $n\ge7$, every connected $K$ of this kind has interior points or separates the plane. Neighboring pieces of $K$ meet in similar copies of a single contact set. In a dendrite exactly one first-level piece meets this set, whereas for $n\ge7$ at least $n-\lceil2\sqrt{n-1}\rceil\ge2$ do, a bound attained by explicit quadratic parameters. The proof combines a planar translation obstruction for replicated contact arcs with an estimate based on Jensen's formula. After normalizing to $z\mapsto t+z/c$, $t\in\{-n+1,-n+3,\ldots,n-1\}$, the connectedness locus is the closure, outside the unit disk, of the roots of monic integer polynomials of height at most $n-1$. The same estimate shows that, for $n\ge7$, every such polynomial with a non-real root $c$ satisfying $|c|^2\ge\lceil2\sqrt{n-1}\rceil+1$ has subleading coefficient of absolute value at most $\lceil2\sqrt{n-1}\rceil$, and that this bound is sharp. Accordingly, for $n\ge7$, at every non-real boundary point $c$ of the locus with $|c|>1$, each expansion $c=\sum_{j\ge0}d_jc^{-j}$ with integer digits $|d_j|\le n-1$ has $|d_0|\le\lceil2\sqrt{n-1}\rceil$.