Ayreena Bakhtawar

Brjuno Functions: Some Interactions Between Number Theory and Holomorphic Dynamics

In the study of the dynamics of holomorphic maps, it is a natural question to ask whether the map is holomorphically linearizable in a neighbourhood of its fixed point. Yoccoz used the Brjuno function to characterize the rotation numbers yielding the linearizability, within the class of a one parameter family of quadratic polynomials, with an indifferent fixed point at zero. He showed that, within this parametric family, the value of the Brjuno function at a certain parameter value is closely linked to the size of the conformal radius of the Siegel disk of the corresponding map.
This talk will revisit both classic and recent results on Brjuno-type functions, focusing on their local and global properties, including their minima, as well as other key characteristics. We will also explore these functions through extensions of the continued fraction algorithm, shedding light on their behavior and relevance within dynamical systems