Geometry, arithmetic, and dynamics of S-adic systems II
speaker: Jörg Thuswaldner (University of Leoben)abstract: Properties of substitutive symbolic dynamical systems under the Pisot hypothesis tend now to be well understood. They are conjectured to have pure discrete spectrum. The aim of these two lectures is first to recall this conjecture (the Pisot conjecture), but also to discuss current extensions of substitutive systems to nonalgebraic parameters using the so-called S-adic framework. We illustrate these extensions with expansions issued from multidimensional continued fraction algorithms, which yields in particular explicit symbolic codings for a.e. toral translation on the two-dimensional torus. We then discuss how to apply these results to produce bounded remainder sets for Kronecker maps.
With the S-adic systems we associate linear Anosov mapping families in a natural way. Defining nonstationary Markov partitions we are able to provide symbolic models of these systems by using nonstationary subshifts of finite type. The pieces of the corresponding generating nonstationary Markov partitions are fractal sets which are S-adic generalizations of the classical Rauzy fractals. These fractals, which are fundamental domains of lattices in the n-dimensional real vector space can be used to relate multidimensional continued fractions to cross sections of the Weyl chamber flow. As an example we get linear Anosov families on the 3-torus associated with the Brun continued fraction algorithm. These are hyperbolic biinfinite sequences of toral automorphisms provided by the Brun matrices. We strongly use here the fact that the Brun renormalization cocycle is Pisot.
This is joint work with P. Arnoux, M. Minervino, and W. Steiner.
timetable:
Thu 7 Apr, 14:45 - 15:30, Aula Dini
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