2024/10/02 by Thomas Garrity, Thomas F. Garrity, Otto Vaughn Osterman +2 · 1 citation
Chemistry · Mathematics · #Advanced Mathematical Theories #Algorithm #Chemistry #Computer science #Fraction (chemistry) #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations
paper · pdf · doi:10.48550/arxiv.2410.02032
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2024/10/02 · openalex created_date 2024/10/30 · openalex updated_date 2026/07/28
We study the complexity of S-adic sequences corresponding to a family of 216 multidimensional continued fractions maps, called Triangle Partition maps (TRIP maps), with an emphasis on those with low upper bounds on complexity. Our main result is to prove that the complexity of S-adic sequences corresponding to the triangle map (called the (e,e,e)-TRIP map in this paper) has upper bound at most 3n. Our second main result is to prove an upper bound of 2n+1 on complexity for another TRIP map. We discuss a dynamical phenomenon, which we call ``hidden \R2 behavior,'' that occurs in this map and its relationship to complexity. Combining this with previously known results and a list of counter-examples, we provide a complete list of the TRIP maps which have upper bounds on complexity of at most 3n, except for one remaining case for which we conjecture such an upper bound to hold.