School on Hyperbolic Dynamics

# seminar: Sums and products of Cantor sets and separable two-dimensional quasicrystal models.

speaker: Yuki Takahashi (University of California, Irvine)

abstract: Arithmetic sums of two Cantor sets appear naturally in dynamical systems, number theory, and also spectral theory. Indeed, the spectrum of the tridiagonal square Fibonacci Hamiltonians, which is a two-dimensional quasicrystal model, is given by sums of two Cantor sets. We show the existence of an open set of parameters which yield mixed interval-Cantor spectra (i.e. spectra containing an interval as well as a Cantor set). On the other hand, the spectrum of the Labyrinth model, which is another two-dimensional quasicrystal model, is given by products of two Cantor sets. We give the optimal estimates in terms of thickness that guarantee that products of two Cantor sets is an interval, and apply this result to show that the spectrum of the Labyrinth model is an interval for sufficiently small coupling constants. We also consider sums of homogeneous Cantor sets, and show that for any two homogeneous Cantor sets with sum of Hausdorff dimensions exceeding 1, one can create an interval in the sumset by applying arbitrary small perturbations to the expanding maps (without leaving the class of homogeneous Cantor sets, but possibly refining the corresponding Markov partition).

timetable:
Thu 15 Jun, 16:00 - 16:20, Aula Dini
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