Research interests

• Fractal geometry, dynamics and number theory. Recently I've been working on problems of fractal geometry and connections with number theory. One motivating conjecture is the idea that if you represent a real number in base 2 and base 3, say, then unless the number is rational (in which case both representationas are eventually periodic), then it is not possible for nboth representations to be too simple, i.e. if one is simple, the other must be complicated. A quantitative formulation was given by Furstenberg in his 1970 paper: he predicted that for irrational t in [0,1], the orbit closure X of t under the map x -> 2x mod 1, and the orbit closure Y of t under x -> 3x mod 1, must satisfy that dim(X')+dim(Y') is at least 1. Here X',Y' are the closures of X,Y respectively.
  
  Nobody is even close to proving (or disproving) this, but there are related, weaker conjectures on which we have been making progress. Pablo Shmerkin and I recently proved a related conjecture of Furstenberg's: if one takes two sets X',Y' as above then their sumset X'+Y' satisfies dim(X'+Y') >= dim(X')+dim(Y') or 1, whichever is smaller. We actually proved something more general, involving invariant measures.
  
  I am currently working on some related problems in fractal geometry and number theory.


• Multidimensional symbolic dynamics. I have been working on the theory of multidimensional shifts of finite type and sofic shifts. An improtant tool here is recursion theory. It has become clear recently that many dynamical properties can be described using recursive measures of complexity. For example, Tom Meyerovitch and I have characterized the numbers that are entropies of SFTs using their recursive properties. One can (almost) characterize thesubactions (directional dynamics) of SFTs via the complexity of the action. Although we have a good understanding of some aspect of the dynamics, though, there still remain many open problems in this field.


• What is the behavior of a 'typical' dynamical system? For Z-actions a lot is known. What dominates in this case is a certain persistence of periodic factors; but when one passes to aperiodic systems a fairly manageable picture emerges. In contrast, the space of Z^d-actions has barely been studied and is far more complex. In place of persistence of periodic factors one has persistence of certain systems related to SFTs, so recursive invariants again come into play. I have been looking into this recently and obtained some preliminary results. For example, the computable Z^2-actions on the Cantor set are dense, and there are actions with dense isomorphism class (weak Rohlin property); but none of the actions with dense isomorphism class are computable. Also, evey isomorphism class is meager (in contrast, Kechris and rosendal showed that for Z-actions, there is a single generic action!).


• Entropy, both topological and measure theoretic, and the relation between them. Entropy is one of the most robust and useful invariants in dynamics, and has an enormous literature, but there are still many interesting questions, such as determining the entropies of some natural classes of systems. Entropy is closely related to dimension, and plays a role in may recent results in number theory, such as the Katok-Einsiendler-Lindenstrauss theorem on the set of exceptions to Littlewood's conjecture. There are also some very interesting new developments, such as Lewis Bowen's recent definition of entropy for free groups.


• Ergodic theorems. I've recently become interested in ergodic theorems for non-singular actions. Following some work by Feldman I obtained an ergodic theorem for finite-rank abelian actions, but the non-abelian case is still a challenging and wide open problem. I'm also interested in effective proofs of ergodic theorems and other convergence theorems.


• The relation between the geometry of a group and its actions. "Finite dimensionality" plays an important role in many ergodic theorems. For some, amenability is a good substitute; there are many cases where this is not good enough, such as the non-singular theorems mentioned above. I am interested in the way geometry effects convergence of ergodic averages; for instance, certain effective versions of ergodic theorems are only known for Z-actions of for abelian actions, and similarly certain entropy estimators work only in those contexts. Another example is the maximal inequality for nonsingular actions, which I recently showed is equivalent to a form of the Besicovitch covering lemma.

For more details try my research-statement, and here are some open problems that interest me.

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