Attacking Classical Problems in Dynamical Systems with Descriptive Set Theory

Abstract:

In his classical 1932 paper, von Neumann asked 3 questions: Can you classify the statistical behavior of differentiable systems? Are there systems where time-forward is not isomorphic to time-backward? Is every abstract statistical system isomorphic to a differentiable system? These questions can be addressed with some surprising consequences by embedding them in Polish Spaces. Indeed the tools answer other questions from the 60’s and 70’s such as the existence of diffeomorphisms with arbitrary Choquet simplexes of invariant measures. Moreover there are surprising analogues to Hilbert’s 10th problem.

In a different category, building on work of Poincare, Smale proposed classifying the qualitative behavior of differentiable systems on compact manifolds. His 1967 paper explicitly argued that the equivalence relation of “conjugacy up to homeomorphism” captures this notion and he proposes classifying it. Call this notion topological equivalence. Very recent joint results with A. Gorodetski show:

• The equivalence relation $E _ 0$ is Borel reducible to topological equivalence of diffeomorphisms of any smooth 2-manifold.

• The equivalent relation of Graph Isomorphism is Borel reducible to topological equivalence of diffeomorphisms of any smooth manifold of dimension 5 or above.

As corollaries, none of the classical numerical invariants such as entropy, rates of growth of periodic points and so forth, can classify diffeomorphisms of 2-manifolds, and there is no Borel classification at all of diffeomorphisms of 5-manifolds.

In the same 1967 paper Smale asks (in different language) whether there is a generic class that can be classified. This is still an open problem.

This talk is hosted by UCLA. Link to the talk site.