Rigidity and Small Divisors in Holomorphic Dynamics

The simplest non-linear systems are driven by quadratic polynomials. That is, "time n" of a state is determined by a quadratic polynomial of "time n-1" of that state. However, despite over a century of intense study, the dynamical features of even quadratic formulae remain far from well understood. For example, complex quadratic polynomials with "small divisors", which may be used to model resonance phenomena, still exhibit mysterious behaviour in many cases.

There has been extensive research on the dynamics of quadratic polynomials over the last three decades. Often, sophisticated tools from different disciplines of mathematics are needed to describe the fine dynamical features of these maps. Usually, a set of such tools is introduced to study the dynamics of a type of quadratic maps, but leads to the successful study of non-linear systems of that type. Thus, an effective set of tools for the study of quadratic polynomials provide the basis of extensive research in the wider area of non-linear systems.

In this project, I develop a new set of tools from different disciplines of mathematics to provide a comprehensive description of the dynamics of certain types of quadratic polynomials. This develops effective techniques from analysis, geometry, and more sophisticated mathematical machinery such as renormalisation and Teichmuller theory.
I will achieve the following major goals.

(1) Small divisors:
A main goal of this research is to introduce a systematic approach to obtain a comprehensive understanding of the dynamics of quadratic polynomials with small divisors. This provides the first examples of such systems with unstable behavior at the center of resonance, whose dynamical behaviour is completely understood.

The Julia set of a quadratic polynomial is the unstable locus of its dynamics. A recent remarkable result of X. Buff and A. Cheritat states that there are quadratic polynomials with small divisors which have observable (positive area) Julia sets. A central problem in the presence of small divisors is to determine arithmetic conditions on the rotation number that leads to observable Julia sets. The proposed research makes major advances on this problem.

(2) Rigidity and density of Hyperbolicity:
The quadratic polynomials that exhibit a certain well understood dynamical behaviour are called hyperbolic. There is a remarkable property, anticipated by P. Fatou in 1920's, stating that any quadratic polynomial may be perturbed to a nearby one with hyperbolic behaviour (by small changes in coefficients in an appropriate normalisation).

The project studies some deep analytic properties of a renormalisation technique to confirm this conjecture for certain types of quadratic polynomials (a Cantor set of parameters). This programme suggests a refined quantitative (in spirit of continued fractions) version of this conjecture to hold.

(3) Generalized Feigenbaum maps:
Period doubling bifurcation is a remarkable phenomenon that appears in the family of quadratic polynomials with real coefficients. There is a wide range of analogous, but more complicated, phenomena that occur when one considers quadratic polynomials with complex coefficients. This reflects the complicated structure of the Mandelbrot set. The dynamical features of such maps with real coefficients have been deeply studied in a period of intense research in 1980's and 90's, while the ones with complex coefficients are largely unexplored. The research proposal uses renormalisation techniques and develops innovative analytical methods to present a detailed description of the dynamics of such a map near degenerate bifurcations.

I will carry out some parts of this major project in collaboration with the leading experts of holomorphic dynamics: A. Avila (Rio, Brazil and Paris, France), X. Buff (Toulouse, France), A. Cheritat (Bordeaux, France), and M. Shishikura (Kyoto, Japan).

The field of Dynamical Systems plays a central role in the development of Mathematics and Physics. It is widely applied in many disciplines to study long term behaviour of environmental, economic, and social systems, such as predicting average values of observables over long periods of time. Thus, the impacts of advances in dynamical systems on everyday life is very important.

The proposed project concerns foundational work to make breakthroughs on central conjectures of Dynamical Systems. It introduces and develops techniques from different disciplines of mathematics such as analysis, geometry, and Diophantine approximation, to provide a comprehensive understanding of highly complicated dynamical behaviours. The flexibility of the methods developed in this programme are very likely to be utilised in a wide range of applications. In particular, the proposed research impacts the areas listed below.

(1) The techniques developed in this research can be used to describe the dynamics of systems with resonances. Resonances are prevalent phenomena in almost periodic events, from the rising of the sun each day to more complicated electromagnetic waves. They often lead to mysterious behaviours. Many such systems, even when given by simple formulae like quadratic polynomials, have remain far from understood to date. One of the main aims of this project is to introduce a systematic approach to successfully study such systems modeled by quadratic formulae on the complex plane. The project answers questions such as whether the set of unstable states are observable (have non-zero probability of occurring).

(2)The project introduces cost effective algorithms for simulating highly complicated non-linear systems. These are dynamical systems arising from complicated bifurcation patterns. Through the developments of this project, I plan to develop software for simulating such systems and making them widely accessible through the internet. I plan to deliver lectures addressing the general public to share the excitements of these ideas and the challenges involved.

(3)The project introduces effective methods that immediately impact many areas such as shape analysis and medical imaging (in healthcare industries), electrical impedance tomography, analysis of water waves. One of the main building blocks of the proposed project is to develop effective analytic methods to describe fine geometric features of the solutions of non-linear partial differential equations, and to obtain optimal estimates on the dependence of the solution of such equations on the data. These methods can be used to establish estimates on conformal mappings and in conformal geometry which have found wide ranges of applications listed above.