My main field of study is Algebraic Dynamics, the study of dynamical systems on algebraic sets. In particular, I study the action of the Mapping Class Group of a compact, orientable surface on the G-Character Variety of the surface, where G is an algebraic group (a matrix group). Both the group and the space in these actions are challenging to explore, and not fully understood. And the resulting dynamical system of these actions contains information on both of these important objects.
Piecewise Contractions: Dynamical systems involving discontinuous maps, even on an interval, are not often or well studied. And even in mild situation can lead to interesting results. Contraction maps in dynamics are considered simple systems. Maps with a single jump discontinuity can also behave like contractions, but with more complicated orbit structure. We call these piecewise contractions. In the paper below, we have used these to model Neuron firings, where stable periodic activity has been observed but not fully understood.
Jimenez, N., Brown, R., et.al., "Locally contractive dynamics in generalized integrate-and-fire neurons", SIAM J. Appl. Dyn. Syst. (2013), 12, no. 3, 1474-1514.
Algebraic Entropy: One property of a dynamical system that measures the complexity of an action is its topological entropy. When a dynamical systems has on a compact space has positive topological entropy, the dynamical system is quite complicated, as nearby orbits tend away form each other exponentially. When the action is algebraic on some affine space, there is a related measure of orbit complexity called algebraic entropy. In simple cases, one can tell by the degree of the map (the maximum degree of the coordinate polynomials in a transformation) what the algebriac entropy is. We study particular cases of mapping class actions on some special linear character varieties to calculate this algebraic entropy.
"The algebriac entropy of the special linear character automorphisms of a free group on two generators", Trans. AMS. 359 (2007), no. 4, 1445-1470.
"The polynomial degree of the special linear characters of a free group on two generators", Geom. Dedicata. (2009), 142, 173–190.
Polynomial Automorphisms: Embed the character variety into affine space, and mapping classes acting on the variety look like affine transformations (polynomial automorphisms) on the variety. However, they do not, in general, lift to actual affine transformations (they are not well-defined). But the embedding does provide global coordinates on the character variety. This allows us to treat cyclic subgroups of mapping class groups acting on character varieties as dynamical systems. Moreover, they act as measure-preserving transformations. This provides the proper playground for future dynamical studies of these actions.
"Automorphisms of the Fricke characters of free groups", preprint..
"Anosov mapping class actions on the SU(2)-representation variety of a punctured torus", Ergod. Th. & Dynam. Sys. 18 (1998), 539-554.