Multidimensional continued fractions and symbolic codings of toral translations

Abstract: It has been a long standing problem to find good symbolic codings for Kronecker toral translations that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties. We construct such codings in terms of multidimensional continued fraction algorithms that are realized by sequences of substitutions. In particular, given any strongly convergent continued fraction algorithm, these sequences lead to renormalization schemes which produce symbolic codings and bounded remainder sets at all scales in a natural way. Such sets provide particularly strong convergence properties of ergodic sums, and are also closely related to the notion of balance in word combinatorics. As strong convergence of a continued fraction algorithm results in a Pisot type property, our approach provides a systematic way to confirm purely discrete spectrum results for wide classes of substitutions. This is joint work with W. Steiner and J. Thuswaldner.

dynamical systems

Audience: researchers in the topic

Bremen Online Dynamics Seminar

Series comments: Talks are approx. 55 min plus discussion. Talks are not recorded.

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