Some Research Interests


My main area of research is ergodic theory and dynamical systems and its applications to other areas of mathematics (for example, geometry, number theory, and analysis). In general terms, dynamical systems describes the long term behaviour of iterating a map on a space. Ergodic theory can be thought of as understanding the behaviour of typical orbits. These simple principles lead to a rich diversity of applications.

 

Lyapunov exponents

we consider the size of a matrix given by multiplying together randomly chosen square matrices (from a finite family). The Lyapunov exponent is the rate of growth of the norm of a typical product, and its existance is guarenteed by a classical result of Furstenberg and Kesten.

In general, it is notoriously difficult to find useful expressions for the Lyapunov exponent which, in particular, allow accurate estimates on its numerical value. However, in the case of positive matrices techniques from Thermodynamic formalism can be used to compute the value to considerable accuracy.

For example, in the cases of the matrices
               (2   1)     and    (3   1)
              (1   1)                (2   1)
chosen according to the usual (1/2, 1/2)-Bernoulli measure, the Lyapunov exponent is

1.1433110351029492458432518536555882994025 ...

 


Linear actions in the Plane

Consider a discrete group G of 2 x 2 real matrices with determinant 1 and fix a non-zero point (x,y) in the plane. It is interesting to look at the orbit G(x,y) = {A(x,y) : A in G} of (x,y) under the natural linear action 

A: (x,y) -> (A[1,1]x + A[1,2]y, A[2,1]x + A[2,2]y). 

If G is a cocompact group then the orbit G(x,y) is dense and the action is ergodic, as was shown by Hedlund (1936). Ledrappier (1997) extended the ergodicity result to groups G? < G with G/G? an infinite abelian group. A simple example is the commutator subgroup G? = [G,G]. 

Francois Ledrappier and I have generalized these results to 2 x 2 real matrices with entries in the complex numbers, quarternions or the very general Clifford numbers. 
 

Whereas for real matrices the problem can be reduced to the dynamics of geodesic and horocycle flows on surfaces, the generalizations come from a study of frame flows, and the associated strong stable foliations. 
Part of the orbit of a cocompact triangle group 
acting on the real plane 


 

Fractals 
and 
Hausdorff 
Dimension 
Fractal dust: 
The Julia set for z -> z? - 3/2 + 2i/3; 
 A quasi-circle: 
The Julia set for z -> z? + i/4

Given a closed set in the plane we can associate to it its Hausdorff dimension, generalizing the usual notion of dimension. Many interesting sets arise as invariant sets for very simple transformations:

Such sets also arise in number theory:

Geometry, 
Number Theory 
and Dynamical Systems

How can we count geoedesics on a surface? A nice solution lies in considering the associated geodesic flow, then closed orbits for the flow correspond to closed geodesics and we have a dynamical problem which we can solve - by using ideas from number theory.  Consider a compact surface with negative curvature. If we take any closed curve on the surface the draw it tight to its shortest length then we have a closed geodesics. It is easy to show there is a countable infinitely of closed geodesics, but we would like to ask how their lengths are distributed.  We can take our lead from the analogous problem in number theory.
 
Counting prime numbers  The number of prime numbers 2,3,5,7,11, ... less than x [let's call it P(x), say] grows like x/log x as x tends to infinity. 
 
This is known as the PRIME NUMBER THEOREM. 
 Counting closed geodesics The number of closed geodesics  N(T)whose length is less than T grows like exp(hT)/hT as T tends to infinity. 
 
(Margulis, Parry-Pollicott). 
Error terms in counting primes  It is natural to ask: Are there better estimates on the number of primes P(x) less than x? The conjecture is that P(x)- li(x) =   O(x 1/2+ e)  for any e > 0.
 
This known as the Riemann Hypothesis
 Error terms in counting closed geodesics One can estimate how quickly [N(T)-li(ehT )]/ ehT tends to zero - exponentially fast!
 
(Dolgopiat, Pollicott-Sharp) 

 
Sums of squares  The number of sums of pairs of 
squares 2, 4, 5, ..., u? + v? less than x grows 
like x/(log x) 1/2 as x tends to infinity. 
 
This is known as  THE HARDY-RAMANUJAN THEOREM 
 Closed geodesic in homology  The number N(T) of closed geodesics whose length is less than T and lie in  the zero homology class has an asymptotic expansion with leading term  Cexp(hT)/Td/2+1 as T tends to infinity. 
 
(Pollicott-Sharp) .

Frame flows, 
compact group extensions 
and skew products

 
A frame flow is a natural generalization of a geodesic flow on a negatively curved manifold..

Instead of simply translating  unit tangent vectors along geodesics, one also translates a whole frame of orthonormal unit tangent vectors  along the same geodesic. This can be viewed as an extension of the geodesic flow by the compact group SO(n-1). 

Although not all such frame flows are ergodic, Brin, Gromov and Karcher showed in the 1980s that ergodicity holds for manifolds close to constant curvature (except in dimensions 7 and 8)

Keith Burns and I extended these results to the two missing dimensions - and also showed the stronger property of stable ergodicity.

Lattice Models, 
Schelling Evolution 
and Dynamical Systems

 
The original disordered system
 
 

The first figure shows a finite lattice with half of the smaller squares shaded ``blue'' and the other half shaded ''mauve'' in a fairly arbitrary way.  Consider the following simple transformation:
  1. Choose two sites (with different colours) from the lattice;
  2. If the blue site has fewer blue nearest neighbours than the mauve site (and the mauve site has fewer mauve nearest neighbours than the blue site) then swap the colours on these two sites. 
Repeating this process, we eventually arrive at the far more more ordered configuration in the second figure.

Alhough this looks like a cellular automaton - it also needs to preserve the total number of sites.  For large lattices it may be possible to describe this using percolation theory.

For small lattices, Howie Weiss and I studied the dynamics of this evolution using simple ideas from dynamical systems.

 

Rates of mixing 
and resonances

A feature of ergodic systems is how fast they approach equilibria - or how fast the system mixes. It is well known that discrete uniformly hyperbolic or ``chaotic'' maps mix exponentially fast. For flows, the problem is more subtle.

Assume that we have a flow f t: M -> M and and invariant measure m.
Given two smooth test functions F,G: M -> R with zero integrals <F> = <G> = 0, then we want to understand the behaviour of the correlation function c(t) = <F(ftx),G>, as t tends to infinity.

The key to studying this problem is take the Fourier transform c^(s) = <c(t), e ist >, and showing that
it can be extended to strip |Im(s)| < e (Pollicott-Ruelle). The poles of the Fourier transform are the resonances. It is their location which determines the speed at which c(t) tends to zero.

Resonances were given a wider audience when Prigogine used these ideas in his description of time arrow of time - relating to the non-reversibility of processes in physics, and other settings.

The connection between periodic orbits for the flow and the ratoe mixing comes the close connection between this Fourier transform c^(s) and dynamical zeta functions (a dynamical zeta function for the flow similar to the famous Riemann zeta function for primes)  (Pollicott)