Abstract. In topological dynamics, one considers the continuous actions of a topological group on a compact space. We will be most interested in minimal actions, those for which every orbit is dense, and Polish groups, groups whose underlying topology is separable and completely metrizable. For countable discrete groups, we show in a precise sense that minimal actions can be wildly complicated. By contrast, for automorphism groups of countable structures, the work of Kechris, Pestov, and Todorcevic establishes a connection between groups whose minimal actions are simple and theorems in finite Ramsey theory. We show that this is in fact the only reason an automorphism group can have well-behaved minimal actions. With this precise correspondence in mind, we then consider possible dynamical formulations of infinite Ramsey theorems.

Abstract. Martin's conjecture is an attempt to make precise the idea that the only natural functions on the Turing degrees are the constant functions, the identity, and transfinite iterates of the Turing jump. The conjecture is typically divided into two parts. Very roughly, the first part states that every natural function on the Turing degrees is either eventually constant or eventually increasing and the second part states that the natural functions which are increasing form a well-order under eventual domination, where the successor operation in this well-order is the Turing jump.

In joint work with Benny Siskind, we prove part 1 of Martin's conjecture for a class of functions that we call measure-preserving. This has a couple of consequences. First, it allows us to connect part 1 of Martin's conjecture to the structure of ultrafilters on the Turing degrees. Second, we also show that every order-preserving function on the Turing degrees is either eventually constant or measure preserving and therefore part 1 of Martin's conjecture holds for order-preserving functions. This complements a result of Slaman and Steel from the 1980s showing that part 2 of Martin's conjecture holds for order-preserving Borel functions.