CSDC Seminar - Brjuno Functions: Some Interactions Between Number Theory and Holomorphic Dynamics (Ayreena Bakhtawar)

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, in the class of all quadratic polynomials. He showed that the Brjuno function is an interpolation of the size of the maximal disk with origin at the fixed point on which the map is linearizable. Moreover, he introduced a bounded holomorphic function measuring the size of the domain of linearizability.

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 generalization of the continued fraction algorithm, shedding light on their behavior and relevance within dynamical systems.