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Large Deviations and Measure Rigidity in Dynamics

Periodic Reporting for period 1 - LDMRD (Large Deviations and Measure Rigidity in Dynamics)

Reporting period: 2015-09-01 to 2017-08-31

The aim of the project was to develop new tools in ergodic theory and dynamical systems, and explore applications to problems related to mathematical physics, geometry and arithmetics. We made substantial progress towards these goals, discovered new mechanisms in the objectives that have recently led to new connections in particular to Fourier analysis and quantum mechanics.
In our first objective we wanted to advance large deviation theory for noncompact dynamical systems, which is deeply connected to Diophantine approximation, and homogeneous dynamics such as the Teichmuller flow on translation surfaces. Our plan was to deduce new subexponential large deviation bounds for Gibbs measures on the countable Markov shift and explore how these results are linked to applications such as Pomeau-Manneville dynamics describing intermittence in the theory of turbulent flows, dynamical properties of the Gauss map. We made substantial progress towards these objectives, discovered some subtle challenges on proving large deviations in nonuniformly hyperbolic setting and we have various articles in preparation on these projects. In the second objective we wanted to investigate Host-type measure rigidity theory for toral automorphisms and homogeneous dynamics, which relates to currently ongoing research on measure classification theorems, which have been influential in several applications such Diophantine approximation and quantum ergodicity. We in particular discovered key challenges of the Host type measure rigidity theory and as a result we established a new breakthrough in quantum ergodicity.
For the first objective, using recently developed tools from probability theory and stochastics, we were able to find necessary conditions on potentials and observables that give rise to subexponential large deviations and currently we are in progress of translating the probablistic ideas from heavy tail processes to obtain dynamical language to obtain a full characterisation polynomial large deviations. On the second objective, we found methods to deal with the low dimensional tori through reduction to the one dimensional situation earlier dealt by Hochman and Shmerkin. Moreover, we explored the connection between measure rigidity to quantum chaos and established a new direction in quantum chaos, where we replace the semiclassical limit by the “thermodynamic limit” by adapting the coarse geometric notions of Gromov-Hausdorff or Benjamini-Schramm
convergence inspired by statistical mechanics (for example, the large scale limits of random processes on graphs giving out SLE-curves). This latter part will have wide impact throughout the communities working in measure rigidity and quantum chaos, number theory and geometry and topology.
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