"Work performed mostly focused on: (1) renormalization of multicriticical circle maps, (2) renormalization of Lorenz maps, and (3) the dynamics of algorithms in reinforcement learning.
1. This work is in collaboration with Prof. Pablo Guarino at UFF, Brazil. Current methods for analyzing critical circle maps rely on complex methods that do not easily extend to arbitrary numbers of critical points and arbitrary exponents. Our research investigated how the methods of M. Martens for unimodal maps could be adapted to the multicritical circle case. This is still work in progress.
2. To investigate the possible renormalization behavior for Lorenz map a numerical experiment was carried out. Current methods are not able to rigorously prove the behavior of these systems so a conjecture was formulated based on the result of the numerics. This is published in the article ""The Lorenz Renormalization Conjecture"". It was found that the behavior of these operators is much more intricate than previously thought. This research was presented at Warwick University, UK, UNSW, Australia, and Inishmore, Ireland.
3. Reinformcement learning algorithms seek to train an agent to learn about its environment by maximizing future expected returns. Together with S. van Strien the learning problem for two agents playing an iterated prisoners dilemma game was investigated. Previous research indicate that these algorithms may exhibit chaotic behavior but this was never proven rigorously. We started applying dynamical systems techniques to understand whether this is truly the case or not. An article is under preparation."
