Dynamical Systems Learning Seminar

Adam Śpiewak:

It is known since a paper by Kifer, Peres and Weiss that dim(μ) is uniformly bounded away from 1 among all probability measures μ on the unit interval which make the digits of the continued fraction expansion i.i.d random variables. I will present a recent paper Maximising Bernoulli measures and dimension gaps for countable branched systems by Simon Baker and Natalia Jurga, where the authors prove that there exists a measure maximising the dimension in this class. The results are valid for a more general class of countable branched systems.

Adam Abrams:

For real continued fractions, the natural extension of the Gauss map has a global attractor with "finite rectangular structure" owing to the "cycle property" satisfied by each algorithm. In this talk I will present a new property satisfied by a large variety of complex continued fraction algorithms and use it to explore the structure of bijectivity domains for natural extensions of complex Gauss maps. These domains can be given as a finite union of Cartesian products in $\mathbb C\times\mathbb C$, where in one complex coordinate the sets come from explicit manipulation of the algorithm, and in the other coordinate the sets are determined by experimental means.

Artem Dudko: