A Rauzy fractal unbounded in all directions of the plane
M\'elodie Andrieu
TL;DR
This paper constructs a specific Arnoux-Rauzy word demonstrating that its Rauzy fractal is unbounded in all directions, with an infinite imbalance, revealing new geometric properties of such words.
Contribution
It introduces an Arnoux-Rauzy word with a difference set equal to A9^3, showing the Rauzy fractal is unbounded in all directions, a novel geometric insight.
Findings
Difference set equals A9^3
Rauzy fractal is unbounded in all directions
The word exhibits infinite imbalance
Abstract
We construct an Arnoux-Rauzy word for which the set of all differences of two abelianized factors is equal to . In particular, the imbalance of this word is infinite - and its Rauzy fractal is unbounded in all directions of the plane.
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Taxonomy
Topicssemigroups and automata theory · Mathematical Dynamics and Fractals · Geometric and Algebraic Topology