## Final Report Summary - CHAPARDYN (Chaos in Parabolic Dynamics: Mixing, Rigidity, Spectra)

The project focused on 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 shed light on the chaotic 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 mechanisms that generate chaos in parabolic systems, such as the key role played by shearing phenomena (in particular a recentrly introduced new form of quantitative shearing adapted to systems with singularities) in order to show subtle chaotic properties such as mixing and disjointness.

Some of the outcomes also indicate that common features may become evident only when studying perturbations of the classical examples (which are too symmetric and hence display special properties). For example, this seems to be the case for mixing in nilflows (which appears only for non trivial perturbations) and for disjointness properties (such as disjointness of rescalings) which were shown to be typical among a certain class of perturbations (time reparametrizations) of certain parabolic flows (horocycle flow and some locally Hamiltonian flows).

Finally, 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).

The research shed light on the chaotic 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 mechanisms that generate chaos in parabolic systems, such as the key role played by shearing phenomena (in particular a recentrly introduced new form of quantitative shearing adapted to systems with singularities) in order to show subtle chaotic properties such as mixing and disjointness.

Some of the outcomes also indicate that common features may become evident only when studying perturbations of the classical examples (which are too symmetric and hence display special properties). For example, this seems to be the case for mixing in nilflows (which appears only for non trivial perturbations) and for disjointness properties (such as disjointness of rescalings) which were shown to be typical among a certain class of perturbations (time reparametrizations) of certain parabolic flows (horocycle flow and some locally Hamiltonian flows).

Finally, 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).