Rauzy Fractals from PV (Pisot) to Non-PV

Jeudi 26 mars 2009 15:45-16:45 - Harriss Edmund - Imperial College

Résumé : It is well known that hyperbolic automorphisms of the 2-torus admit a Markov partition. This markov partition is closely related to 1 dimensional tilings of the line called Sturmian tilings. In the general case any hyperbolic toral automorphism of the d-torus admits a Markov partition, however a result of Bowen shows that, if $d > 2$ and the automorphism is algebraically irreducible, the elements of the Markov partition can never have a smooth boundary : this boundary is a non-trivial fractal set, and is difficult to construct in the general case.
In 1982 Gerald Rauzy found a method of constructing the Markov partition of an irreducible hyperbolic automorphism of the 3-torus using a substitution rule on three letters.
This work has been generalised to construct the Markov partition for many (conjectorally all) irreducible PV toral automorphisms.
Recent work of Arnoux, Furukado, Ito and H has opened up this geometric construction in the more complicated non-PV case, where there is more than 1 eigenvalue of absolute value greater than 1.
This talk will focus on the geometric construction of Rauzy fractals that give the Markov partitions and the related substitution tilings. Starting with the original example of Rauzy, and the general PV-case before showing how this construction can be generalised to the non-PV case.

Lieu : 425 - 121-123

Rauzy Fractals from PV (Pisot) to Non-PV  Version PDF