The Dixmier-Moeglin equivalence and dynamical properties of automorphisms

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My proposed research deals with interactions between algebra and dynamics. In recent years, it has been noticed that one can associate an algebraic object to a given geometric object along with certain geometric data. These objects are called twisted homogeneous coordinate rings. What is interesting is that many interesting algebraic properties appear to be linked with what are called "dynamical properties". For example, when one has an abstract space and a transformation of this space, one can iterate the transformation and ask what happens to a given point in the space over time as we repeatedly iterate. These sorts of questions are at the heart of dynamics, and understanding some of these questions actually gives us insight into the associated twisted homogeneous coordinate rings.