Seminar 4.19.18 Samuel

Title: A classification of intermediate β-transformations

Speaker: Tony Samuel (Cal Poly San Luis Obispo)

Abstract: In this talk we consider transformations of the unit interval of the form $\beta x + \alpha \bmod{1}$ where $1 < \beta < 2$ and $0 \leq α \leq 2 – \beta$.  These transformations are called intermediate β-transformations.  We will discuss some old and new results concerning these transformations, for instance, their kneading sequences, their absolutely continuous invariant measures and dynamical properties such as topological transitivity and the sub-shift of finite type property.  Moreover, we address how the kneading sequences and absolutely continuous invariant measures change as we let $(\beta, \alpha)$ converge to $(1, \theta)$, for some $\theta \in [0, 1]$. Finally, some open problems and applications of these results to one-dimensional Lorenz maps and quasicrystals will be alluded to.