Objective
The ergodic theory of parabolic dynamical systems is an area in smooth ergodic theory that is relevant for its connections to mathematical physics and to analytic number theory. A dynamical system is parabolic whenever its orbits diverge at an intermediate (often polynomial) rate between bounded/logarithmic (called elliptic) and exponential (called hyperbolic) rate.
The proposal tackles several outstanding questions in the ergodic theory of parabolic flows with emphasis on quantitative aspects. The main goal is to go beyond renormalization techniques that have proved extremely powerful
in several classes of examples: Interval Exchange Transformations and Flows on surfaces, horocycle flows, nilflows on quotients of step-two nilpotent groups and Gauss sums. Renormalization methods are not available in other equally fundamental examples of similar nature: billiards in non-rational polygons, higher step nilflows and non-horospherical unipotent flows in homogeneous dynamics. A unified approach to effective ergodicity is proposed that encompasses all of the above mentioned examples.
Outstanding questions include ergodicity and existence of periodic orbits of
non-rational billiards in polygons, effective ergodicity of higher step nilflows
with optimal deviation exponents and applications to bounds on Weyl sums
for higher degree polynomials and effective ergodicity of non-horospherical unipotent flows on semi-simple finite-volume quotients.
The analytical foundations of the method lie in the study of invariant distributions for parabolic flows. In the examples considered the analysis
can be carried out by methods of non-Abelian Fourier analysis (theory of
unitary representations). In general, for non-homogeneous parabolic flows,
all questions concerning invariant distributions and their relevance for smooth ergodic theory are wide open. Several problems to probe the question of existence of invariant distributions are proposed.
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Keywords
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Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
MAIN PROGRAMME
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Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
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Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
HORIZON-ERC - HORIZON ERC Grants
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Call for proposal
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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2023-ADG
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Net EU financial contribution. The sum of money that the participant receives, deducted by the EU contribution to its linked third party. It considers the distribution of the EU financial contribution between direct beneficiaries of the project and other types of participants, like third-party participants.
95011 Cergy-Pontoise
France
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.