On the Factor Complexity Associated with a Family of Multidimensional Continued Fraction Algorithms

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 $\R^2$ 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.