Multidimensional continued fractions and symbolic codings of toral translations

TL;DR

This paper develops symbolic codings for multidimensional toral translations using continued fraction algorithms, confirming the Pisot conjecture for various substitution sequences and providing new insights into bounded remainder sets.

Contribution

It introduces a systematic method to generate symbolic codings for toral translations via multidimensional continued fractions, confirming the Pisot conjecture for several well-known algorithms.

Findings

Confirmed Pisot conjecture for Jacobi–Perron, Brun, Cassaigne–Selmer, and Arnoux–Rauzy algorithms.

Provided symbolic codings for almost all 2D toral translations with low factor complexity.

Extended symbolic codings to 3D translations with multiscale bounded remainder sets.

Abstract

It has been a long standing problem to find good symbolic codings for translations on the $d$-dimensional torus that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties. Inspired by Rauzy's approach we construct such codings in terms of multidimensional continued fraction algorithms that are realized by sequences of substitutions. In particular, given any exponentially convergent continued fraction algorithm, these sequences lead to renormalization schemes which produce symbolic codings of toral translations and bounded remainder sets at all scales in a natural way. The exponential convergence properties of a continued fraction algorithm can be viewed in terms of a Pisot type condition imposed on an attached symbolic dynamical system. Using this fact, our approach provides a systematic way to confirm purely discrete spectrum results for wide classes of symbolic dynamical systems. Indeed, as our examples illustrate, we are able to confirm the Pisot conjecture for many well-known families of sequences of substitutions. These examples comprise classical algorithms like the Jacobi--Perron, Brun, Cassaigne--Selmer, and Arnoux--Rauzy algorithms. As a consequence, we gain symbolic codings of almost all translations of the $2$-dimensional torus having factor complexity $2n+1$ that are balanced for words, which leads to multiscale bounded remainder sets. Using the Brun algorithm, we also give symbolic codings of almost all $3$-dimensional toral translations having multiscale bounded remainder sets.