Tony Samuel: A meander through the space of intermediate β-transformations

In this talk we consider transformations of the unit interval of the form βx + α mod 1, where 1 < β < 2 and 0 ≤ α ≤ 2 - β. 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 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 (β, α) converge to (1, θ), for some θ ∈ [0, 1]. Finally, some open problems and applications of these results to one-dimensional Lorenz maps and quasicrystals will be alluded to.