## 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) 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.

### Closed orbits and Dynamical Zeta functions

Fifty years ago Selberg and Huber showed that for compact surfaces with constant negative curvature there is a ``Prime Geodesic Theorem'' (analogous to the Prime Number Theorem in number theory) giving a simple and elegant asymptotic formula for the number of closed geodesics N(T) whose length is at most T.

This was subsequently extended to surfaces of variable curvature by Margulis (Fields medallist) in 1969, but without the error term established in the constant curvature case, i.e., there exists h > 0 such that

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.

In another direction, Sarnak-Philips considered the problem of additionally restricting to geodesics in a given homology class a and gave an asymptotic formulae for the number N(a, T)\$ of such closed geodesics on a surface of constant negative curvature whose length is at most T. This again generalizes:

Theorem (P., P.-Sharp): For a surface of (variable) negative curvature and genus g > 1 there exist constants c0, c1, c2 ...such that

N(a,T) = ehT/T g+1 (c0+ c1/T + c2/T2 + ...) as T tends to infinity.

This is analagous to the Hardy-Ramamnujan theorem which states that the number of sums of pairs of squares 2, 4, 5, ..., u 2 + v 2 less than x grows like x/(log x) 1/2 as x 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.

Prime Orbit Theorem (Parry-P.): There exists a constant h > 0 such that P(t) ~ ehT/hT as T tends to infinity

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.

### Linear actions in the Plane

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.

The picture shows part of the orbit of a cocompact triangle group acting on the real plane.

### Decay of Correlations.

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:

Theorem (P.): The Fourier transform F(&rho)(s) = &int exp(its)&rho(t) dt is meromorphic for |Im(s)| < &epsilon .

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.

Ruelle Conjecture (P.) : For any 0 < &delta < 1, the suspension semiflow for a function r(x) = 1 + &delta sin(2 &pi x) over doubling map z -> z 2 on the unit circle mixes exponentially.

By contrast, I also showed that there exist many examples of hyperbolic flows which mix arbitrarily slowly.

### Ergodicity of partially hyperbolic systems.

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.

Theorem (Burns-P.): For each d > 1 there exists &epsilon > 0 such that if the sectional curvatures are &epsilon --pinched then the frame flow is (stability) ergodic.

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).

### Fractals and Hausdorff Dimension

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.

Theorem [Keane-Smorodinsky-Solomyak Conjecture] (P.-Simon) The Hausdorff dimension of

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:

Theorem (Jordan-P.) For fat Sierpinski gaskets there are explicit ranges of l where the dimension has a known value for almost all l. There are other explicit ranges of &lambda where the set can be shown to have positive lebesgue measure for almost all l.

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.

### Explicit Formulae and Algorithms.

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).

Theorem (P.-Rocha) : There is a formula for det(g) in terms of a rapidly convergent series each of whose coefficients is explicitly given in terms of the lengths of finitely many closed geodesics.

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.

Theorem (P.) For a compact surface with curvature &kappa = -1 the Selberg zeta function Z(s) can be written in terms of a convergent series each of whose coefficients depends on s and are explicitly given in terms of the lengths of finitely many closed geodesics.

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).

Theorem (Jenkinson-P.): There exists a very efficient algorithm for computing the Hausdorff dimension of hyperbolic Julia sets using periodic orbit data.

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:

•  Linear Smale horseshoes. In two dimensions the dimension is easy to compute - but not so in higher dimensions, where Howie Weiss and I showed that the Hausdorff dimension of horsehoes can be discontinuous under perturbation.
• Julia set of a rational map.Oliver Jenkinson and I devised an algorithm which computed the dimension of the Juliasets for the examples above to be 09038745968111... and 1.0231991890309691 ... , respectively.

#### Example 2:

Such sets also arise in number theory:

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:

 1 2 1 1

 3 1 2 1
chosen with probability (1/2,1/2) the associated Lyapunov exponent is

&lambda = 1.1433110351029492458432518536555882994025 ...

As a recent application one has the following.

Theorem (P.): There exists a very efficient algorithm for computing the entropy rate for Hidden Markov Chains corresponding to binary symmetric channels.

### Lattice Models,  Schelling Evolution  and Dynamical Systems

 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: Choose two sites (with different colours) from the lattice; 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.