Ergodic theory and dynamical systems

The isomorphism problem in dynamics  dates back to a question of von Neumann from 1932: Is it possible to classify (in some reasonable sense) the ergodic measure-preserving diffeomorphisms of a compact manifold up to isomorphism? We want to study a similar problem: Let C be the Cantor set and let Min(C) stand for the space of all minimal homeomorphisms of the Cantor set. Recall that a homeomorphism f is in Min(C) if every orbit of f is dense in C.  We say that f and g in Min(C) are topologically conjugate if there is a Cantor set homeomorphism h such that f o h = h o g. We prove an anti-classification result showing that even for very liberal interpretations of what a "reasonable'' classification scheme might be, a classification of minimal Cantor set homeomorphism up to topological conjugacy is impossible. We see it as a consequence of the following: we prove that the topological conjugacy relation of Cantor minimal systems TopConj treated as a subset of Min(C)xMin(C) is complete analytic. In particular, TopConj is a non-Borel subset of Min(C)xMin(C). Roughly speaking, it is impossible to tell if two minimal Cantor set homeomorphisms are topologically conjugate using only a countable amount of information and computation.

Our result is proved by applying a Foreman-Rudolph-Weiss-type construction used for an anti-classification theorem for ergodic automorphisms of the Lebesgue space. We find a continuous map F from the space of all trees over non-negative integers with arbitrarily long branches into the class of minimal homeomorphisms of the Cantor set. Furthermore, F is a reduction, which means that a tree T is ill-founded if and only if F(T) is topologically conjugate to its inverse. Since the set of ill-founded trees is a well-known example of a complete analytic set, it is impossible to classify which minimal Cantor set homeomorphisms are  topologically conjugate to their inverses.

This is joint work with Konrad Deka, Felipe García-Ramos, Kosma Kasprzak, Philipp Kunde (all from the Jagiellonian University in Kraków).

Various algebraic phenomenons in homogeneous dynamics have non algebraic counterparts. For example, the equidistribution of Hecke neighbors can be seen as a non-algebraic counterpart of Duke's Theorem. In this talk, we consider non-algebraic counterparts of the p-Adic Littlewood Conjecture and of Duke's Theorem for subcollections.
One of the non-algebraic counterparts of the p-Adic Littlewood Conjecture involves unboundedness of the A-orbits of arbitrary choices of p-Hecke neighbors of a lattice as p goes to infinity along the primes.
We prove, using expanders, a bootstrap argument and the equidistribution of Hecke neighbors, that the set of exceptions for this conjecture has Hausdorff dimension strictly smaller than 1 in [0,1] (where we assign lattices to points in [0,1]). Moreover, we discuss evidence for the conjecture in some cases using GRH.

This talk is based on a joint ongoing work with Erez Nesharim from the Technion.