I'm interested in topological and symbolic dynamics, ergodic theory,
entropy and information theory, espcially for group actions, and
recently in fractal geometry and applications to number theory and
dynamics. More specifically,
- 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
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
- 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
For more details try my research-statement, and here are
some open problems that interest
Mike Hochman's home