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CHAPARDYN Report Summary

Project ID: 335989
Funded under: FP7-IDEAS-ERC
Country: United Kingdom

Mid-Term Report Summary - CHAPARDYN (Chaos in Parabolic Dynamics: Mixing, Rigidity, Spectra)

The project main theme is the mathematical investigation of chaos in parabolic dynamics. Parabolic dynamical systems are mathematical models of the many phenomena which display a "slow" form of chaotic evolution behaviour: nearby trajectories diverge, hence making the system hard to predict, but polynomially in time, in contrast with the exponential speed of divergence of the well understood hyperbolic systems.
The research done so far shed light on the caotic features, in particular mixing and spectral properties, of a number of parabolic systems, including examples which arise in mathematical physics (e.g. the light ray dynamics in systems of perfect retroreflectors or billiards in pseudo-integrable systems), in solid state physics (locally Hamiltonian flows on surfaces) and as perturbations of parabolic flows in the algebraic setting (e.g. time changes of nilpotent and unipotent flows).
The results have highlighted common features and mechanisms that generate chaos in parabolic systems, such as the key role played by shearing effects in mixing and spectral results and the genericity of mixing properties. Furthermore, we have developed new methodologies which sits at the interface between several areas of research, in particular Teichmueller dynamics, spectral theory and infinite ergodic theory and involve a mixture of analytical, geometrical and combinatorial tools.
In addition to the above mentioned applications to systems which arise in mathematical physics, we have also explored connections between parabolic dynamics and number theory (examples include results on generalized Lagrange spectra and gap distributions) and probability theory (distributional limit theorems).

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United Kingdom
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