This project investigates aspects of low dimensional dynamical systems. The study of such transformations, especially area preserving ones, has a long history going back to Poincare and G. D. Birkhoff. In particular the project considers smooth group actions on surfaces and the relation between the algebraic properties of the group and the dynamics the action exhibits. A related question addressed by this proposal is the question of the existence of global fixed points for two-dimensional dynamical systems and how this relates to the algebraic properties of the system.
This project investigates aspects of dynamical systems on surfaces. There are numerous applications of results in this area to broader fields of science, especially to classical mechanics and more modern chaos theory. The novelty of the proposed research is that it addresses the relationship between the algebraic nature of dynamical systems called ``group actions'' and the geometric or topological behavior they exhibit. Anticipated results from this proposal will advance our knowledge of dynamical systems and will explore new relationships between dynamics and algebra.
