On the Linear Complexity Associated with a Family of Multidimentional Continued Fraction Algorithms

TL;DR

This paper investigates the complexity of sequences generated by multidimensional continued fraction algorithms, establishing upper bounds for specific maps and exploring dynamical phenomena affecting complexity.

Contribution

It proves upper bounds on the complexity of certain TRIP maps and introduces the concept of hidden R^2 behavior in these dynamical systems.

Findings

The (e,e,e)-TRIP map has complexity at most 3n.

Another TRIP map has complexity at most 2n+1.

Identification of a dynamical phenomenon called hidden R^2 behavior.

Abstract

We study the complexity of S-adic sequences corresponding to a family of 216 multi-dimensional 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 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 $R^2$ 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.