Multidimensional continued fractions and symbolic codings of toral translations

Valérie Berthé, Wolfgang Steiner, Jörg Thuswaldner

Research output: Contribution to journalArticleResearchpeer-review


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.
Original languageEnglish
Pages (from-to)4997-5057
Number of pages61
JournalJournal of the European Mathematical Society
Issue number123
Publication statusPublished - 20 Dec 2022

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