Minimal rank of $\beta$-expansions.
speaker: Maria Rita Iacò (TU Graz)abstract: A special focus concerning $beta$-expansions lies on arithmetic conditions which guarantee the finiteness of the expansions. In particular, we will be interested in the so-called positive finiteness property of $\beta$-expansions, stating that each polynomial in base $\beta$ with non-negative integer coefficients has a finite admissible expression in in the corresponding number system. The aim of this talk is to provide a connection between the positive finiteness property of $\beta$-expansions and two properties ensuring pure discrete spectrum of the odometer. The first one is called Hypothesis B and it is a condition on the carries of the digits in the expansion of positive integers in a base system defined by a linear recurrence of order d. The second one is called quotient mapping condition (QM) and it is an algebraic hypothesis introduced in the framework of tilings associated to Pisot-substitutions. We will show that the Hypothesis B is equivalent to the positive finiteness property of $\beta$-expansions and the quotient mapping condition. This is joint work with W. Steiner and R.F. Tichy.
timetable:
Tue 5 Apr, 15:30 - 16:00, Aula Dini
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