seminar: Continued fractions, transducers, and the Serret theorem
speaker: Giovanni Panti (Università di Udine)abstract: The Serret theorem is a basic result in the elementary theory of continued fractions: two real numbers share the same tail in their continued fraction expansions iff they belong to the same orbit under the projective action of PGL(2,Z). Notwithstanding the abundance of continued fraction algorithms in the literature, a uniform treatment of the Serret result seems missing. I will show that there are finitely many possibilities for the subgroups Sigma of PGL(2,Z) generated by the branches of Gauss-type maps, and that each Sigma-equivalence class of reals is partitioned in finitely many tail-equivalence classes. The number of these classes is computable from the finite-state transducers that relate Gauss maps to each other. I will show that these transducers constitute opfibrations of the Schreier graphs of the groups, and relate their synchronizability ---which may or may not hold--- to the a.e. validity of the Serret theorem
timetable:
Wed 14 Mar, 15:00 - 16:00, Sala Conferenze Centro De Giorgi
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