A meander through the space of intermediate $\beta$-transformations

Fractal Geometry and Dynamics

28 November 14:00 - 14:50

Tony Samuel - California Polytechnic State University

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.

Kenneth J. Falconer
University of St Andrews
Maarit Järvenpää
University of Oulu
Antti Kupiainen
University of Helsinki
Francois Ledrappier
University of Notre Dame
Pertti Mattila
University of Helsinki


Maarit Järvenpää


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