Contact multiplicity and a sharp six-map bound for collinear self-similar dendrites
Abstract
Let K⊂ℂ be the attractor of n≥2 similarities z↦λz+bk, 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≤6, and that such dendrites exist for every 2≤ n≤6; no separation or finite-intersection hypothesis is imposed. Hence, for n≥7, 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≥7 at least n−⌈2√(n−1)⌉≥2 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↦ t+z/c, t∈{−n+1,−n+3,…,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≥7, every such polynomial with a non-real root c satisfying |c|2≥⌈2√(n−1)⌉+1 has subleading coefficient of absolute value at most ⌈2√(n−1)⌉, and that this bound is sharp. Accordingly, for n≥7, at every non-real boundary point c of the locus with |c|>1, each expansion c=∑j≥0djc−j with integer digits |dj|≤ n−1 has |d0|≤⌈2√(n−1)⌉.
Related
- Coding multiplicity in self-similar sets on the line
- Beyond Multiplicity Correction: Sharp Retrieval Thresholds for Attention with Correlated Distractors
- Nearly Sharp Uniform Projection Bounds and Superlinear Vitushkin Inequalities for Deterministic Self-Similar Sets
- Four collinear points or six visible points
- An infinite walk in ℕ16, using only unit steps, with no three collinear points
BibTeX
@misc{espigule2026contact,
title = {{Contact multiplicity and a sharp six-map bound for collinear self-similar dendrites}},
author = {Bernat Espigule},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01291},
eprint = {2611.01291},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01291},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Contact multiplicity and a sharp six-map bound for collinear self-similar dendrites
AU - Bernat Espigule
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01291
UR - https://arxiv.org/abs/2611.01291
AB - Let K⊂ℂ be the attractor of n≥2 similarities z↦λz+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≤6, and that such dendrites exist for every 2≤ n≤6; no separation or finite-intersection hypothesis is imposed. Hence, for n≥7, 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≥7 at least n−⌈2√(n−1)⌉≥2 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↦ t+z/c, t∈{−n+1,−n+3,…,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≥7, every such polynomial with a non-real root c satisfying |c|^2≥⌈2√(n−1)⌉+1 has subleading coefficient of absolute value at most ⌈2√(n−1)⌉, and that this bound is sharp. Accordingly, for n≥7, at every non-real boundary point c of the locus with |c|>1, each expansion c=∑_j≥0d_jc^−j with integer digits |d_j|≤ n−1 has |d_0|≤⌈2√(n−1)⌉.
ER -