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)
mathematical physics and mathematical biology. 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.
N(T) ~ ehT /hT, as T tends to infinity.
However, the following stronger result (with an error term) comes from developing a more dynamical approach.Theorem [Prime Geodesic Theorem with error] (P.-Sharp): For surfaces of (variable) negative curvature there exists h > 0 and &epsilon>0 such that
N(T) = Li(ehT) (1+ O(e-&epsilon T), as T tends to infinity.
N(a,T) = ehT/T g+1 (c0+ c1/T + c2/T2 + ...) as T tends to infinity.
Another more subtle result on the distribution of lengths is a pair correlation result comparing lengths of closed geodesics (ordered by their word length) which we proved a few years later.
However, the most general asymptotic result counts the number P(T) = Card{ &lambda(&tau) < T} of closed orbits &tau for weak mixing Axiom A flows (in the sense of Smale), or hyperbolic flows, with least period &lambda(&tau) less than T.
This result is analogous to that of the prime number theorem (i.e., the number of primes less than x is asymptotic to x/log x). This analogy extends to the study of a suitable dynamical zeta function
&zeta (s) = &prod &tau (1- e -s h l (&tau )) -1
defined by analogy with the Riemann zeta function. The dynamical zeta function converges in the half plane Re(s) > 1 and I showed that this always has a meromorphic extension to a strip 1 - &delta < Re(s) < 1 (although not necessarily to the entire complex plane). I also studied the dependence of the entropy h on the flow (or the Riemannian metric, in the case of geodesics flows). The approach which I developed to prove these results is part of what is now generally called ``thermodynamic formalism'' .
I have also developed a parallel theory for Poincare series and orbital counting functions (e.g., the orbit of a point in the universal cover of a negatively curved manifold under the action of the fundamental group). In this setting additional progress was made using ideas from geometric group theory.
I also completed a posthumous paper with Parry which proves a dynamical analogue of Bauer's theorem in algebraic number theory, complementing our earlier version of Chebotarov's theorem and Sunada's construction of isospectral surfaces.
The orbital counting results were used in turn in the proof of the remarkable Babillot-Ledrappier result on horocycle ergodicity for periodic surfaces. I subsequently gave a new proof of, which subsequently lead to my work with Ledrappier on counting problems for more general Clifford groups and p-adic groups.
Consider a discrete group G &sub SL(2, R) 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 &isin G} of (x,y) under the natural linear action
A: (x,y) -> (A1,1 x + A1,2 y, A2,1 x + A2,2 y).
If G is a cocompact group then the orbit G(x,y) &sub R 2 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] &sub G.
In the study of dynamical systems, one of the central themes is their long term statistical behaviour. Of particular interest is the speed at which the system approaches its equilibrium state, which is described by the ``decay of correlations'' (or rate of mixing). More precisely, given a smooth flow &phit: M -> M on a compact manifold M, a suitable f-invariant probability measure m and real valued square integrable functions F,G one can associate the correlation function
&rho(t) = &int F&phit. G dm - &int F dm &int G d m.
The behaviour of this function is particularly important for ``chaotic systems'', one of the principle mathematical models for which are the hyperbolic flows &phit: M -> M (which includes the classical examples of geodesic flows on negatively curved manifolds). The behavior of r(t) (where m is a Gibbs measure for a Holder potential and F,G are Holder functions) is then determined by the following result:
Thus the asymptotic behaviour of t(t) is controlled by the poles in this extension (often now called P.-Ruelle resonances) and this is essentially the only technique known to study the decay of correlations. For example, this result was used by Prigogine (Nobel laureate) in his work on ``The arrow of time''; and by Dolgopyat to prove exponential decay of correlations for geodesic flows on negatively curved surfaces and, by extension, Avila-Goeuzel-Yoccoz to prove the analogous result for Teichmuller flows. It was also used to prove the following key conjecture of Ruelle for a simple model of hyperbolic flows.
By contrast, I also showed that there exist many examples of hyperbolic flows which mix arbitrarily slowly.
A more general class of systems than hyperbolic systems are the partially hyperbolic systems. The canonical example of this is the frame flow associated to a compact d-dimensional manifold M with negative sectional curvatures, by which orthonormal frames of tangent vectors are parallel transported along geodesics (which is a G = SO(d-1)-extension of the associated hyperbolic geodesic flow). The following result completed a programme of Brin and Gromov.
Parry and I also developed a programme to study the stability of ergodicity (and mixing) for skew products over hyperbolic diffeomophisms. This lead to a fairly complete classification of ergodic skew products, in part through the developments in the theory of measurable Livsic theorems. Recently, I have also studied ergodicity problems in other settings and using more probabilistic techniques I have criteria for uniqueness of certain Gibbs measures (or g-measures).
A basic method in studying ``typical'' fractal limit sets associated to a parameterised family of contractions is a technique developed with Simon called transversality. This was originally used to prove the following.
X(&lambda ) := {&sum in &lambda n : in&isin {0,1,3} }
is -log(3)/log(&lambda) for almost all 1/4 < &lambda < 1/3.However, the transversality method subsequently proved to be the main tool in studying many different problems, including Solomyak's well known solution of the Erdos Conjecture. Another basic example of a ``fractal set'' is the Sierpinski gasket defined for a given finite family S &sub {0, ... , d-1} x {0, ... , d-1} by
X(&lambda) : = {&sum in &lambdan, &sum jn &lambda n) : ( in, jn) &isin S}.
When l = 1/d this is a standard Sierpinski gasket, and the dimension is easily computed. However, for 1/d < l < 1 this is called a ``fat Sierpinski gasket'' and one of the few results known is the following:
However, developing the approach in this theorem, lead to resolving an old conjecture (by Peres-Solomyak) showing that there exists a self-affine set of nonzero measure and empty interior.
A continuing theme of my work is finding explicit (and effectively computable) formulae for characteristic values. For example, for a compact Riemann surface (M,g) with negative Euler characteristic one can consider the famous Determinant of the Laplacian det(g).
In particular, this allows one to efficiently compute the numerical value of det(g) for given surfaces. This method gives an explicit expression for the Selberg zeta function
Z(s) = &prod &tau &prod n (1- e(-(s+n) h &lambda (&tau) )
for a surface of constant negative curvature.
This analysis recently motivated Anantharaman-Zelditch to reformulate problems in Quantum Unique Ergodicity, in terms of residues of related complex functions (for which I can now give explicit formulae). In a different direction, I derived a new algorithm for computing for the Hausdorff dimension of hyperbolic Julia sets (and limit sets of Kleinian groups).
In particular, this method improved on an algorithm of McMullen.
Example 1: |
Fractal dust: The Julia set for z -> z 2 - 3/2 + 2i/3; |
A quasi-circle: The Julia set for z --> z 2 + 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:
E2 = {numbers whose continued fraction expansion contains only digits 1 and 2}
This is closely related to the so called Markoff spectrum for diophantine approximation. Oliver Jenkinson and I have an efficient algorithm for calculating dimensions of such sets. In particular, it shows that the dimension of E2 is (to 25 decimal places) 0.5312805062772051416244686 ...In a similar spirit, I have studied the Lyapunov exponents &lambda = &lambda (A,B)$ associated to positive matrices A,B (as studied by Furstenberg and Kesten) 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.
I have derived expliciit (and effectively computable) formulae for &lambda (A,B) in terms of a convergent series whose terms are given in terms of products of finitely many matrices. For example, with the matrices:
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