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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - IChaos (Intermediate Chaos)

Teaser

Summary

The theory of dynamical systems is a mathematical study of evolution in time. Dynamical chaos is a broad paradigm for the study of systems whose long-term behaviour exhibits a highly sensitive dependence on small changes in the system\'s initial conditions. The trajectories of a chaotic dynamical system are in this case most effectively described in probabilistic terms, for example, by comparison with Brownian motion, in stark contrast with orderly quasi-periodic motion of celestial bodies.

The aim of this project is to develop a mathematical theory of intermediate chaos, the regime of transition between order and strong chaos.

A key example is furnished by self-similar dynamical systems that arise, on the one hand, in geometry as partial isometries in one dimension, and, on the other hand, in the theory of formal languages as dynamical systems generated by substitutions. Systems with self-similarity are in particular used to model quasi-crystals.
An important tool for the study of these systems is dynamical renormalization. In simplest terms, renormalizability means that, despite the fact that the complexity of the system grows, if the system is considered over carefully chosen long spans of time, then the level of complexity is the same as when the system is
considered over a short interval of time. This behaviour is in stark contrast with strongly chaotic dynamical systems whose complexity grows exponentially but also with periodic motion whose complexity does not grow at all.

Another key object of study for the project is point processes, systems that model consecutive occurrences of events that cannot be precisely planned, such as, for example, deaths in different neighbourhoods of London (investigated by John Graunt as early as 1662!). Stemming from problems in demography, point processes have since found applications in areas as diverse as queuing theory, nuclear physics and machine learning.
The project specifically focuses on determinantal point processes that appear in the theory of random matrices.
The theory of random matrices originates from the problem of describing the behaviour of heavy nuclei in physics as well as the problem of describing large samples in multivariate statistics.

The overall objective of the project is to give a quantitative description of intermediately chaotic dynamical systems in terms of their dynamical characteristics such as limit theorems and spectrum. The project lies at the intersection of a wide variety of mathematical disciplines, requiring tools, in particular, from classical, complex, harmonic and functionl analysis, geometry, combinatorics,
probability theory and stochastic processes, and is possible due to the broad expertise of the ERC team in various aspects of the theory of dynamical chaos and related geometric, analytic and probabilistic problems.

Work performed

The key results obtained so far are:

1) A detailed description of quasi-symmetries of determinantal point processes, the relationship between Palm measures of point processes. Palm measures are a mathematical tool for describing the probability of occurrence of a new random event provided one has just occurred. Conrad Palm, an engineer for Ericsson in Stockholm, introduced and studied these probabilities in order to minimize telephone communication waiting time.

2) A detailed quantitative description, achieved in collaboration with Boris Solomyak, of the spectrum for a wide class of systems arising in geometry and exhibiting self-similarity. This is a first quantitative description of the spectrum of dynamical systems with intermediate chaos.

3) A detailed geometric and probabilistic description, obtained in collaboration with Y. Qiu, A. Shamov, S. Fan, of reproducing kernels of determinantal point processes.

4) A theorem, proved in collaboration with Andrei Dymov, on approximation of trajectories of the sine-process, a point process playing a key rÃ´le in the study of heavy nuclei, by a generalized stochastic process related to
the Gaussian Free Field.

5) A theorem, proved in collaboration with Y. Qiu, on extrapolation for determinantal point processes. This result is motivated by the problem of recovering a continuous signal from sampling performed at given moments of time.

Final results

Dynamical systems and point processes are used to model a wide variety of phenomena in the world that surrounds us.

The project ICHAOS is a quantitative study of intermediate chaos for dynamical systems. For the purpose of this study the project has developed new tools such as, for example, a general formalism of multiplicative functionals in the study of determinantal point processes, a development of the Kolmogorov 0-1 law for point processes as well as a new local property for Palm kernels of determinantal point processes (in collaboration with Y. Qiu and A. Shamov), a development of the Erdos-Kahane method for the study of the intermediate chaos for self-similar dynamical systems of geometric origin (in collaboration with B. Solomyak).

The novel methodology of the project lies at the crossroads of probability, analysis and geometry.
Specifically, further progress is expected in the important question of sampling in function spaces using realizations of point processes. In the simplest situation, sampling corresponds to determining, with the best possible precision, a continuous signal from its values at given moments of time. Sampling plays a key rÃ´le in signal transmission, and the project team, together with new members who are scheduled to join us in September 2018, is working on developing a new methodology for the sampling of determinantal point processes. Specific progress is also expected on lower bounds for the spectrum of symbolic dynamical systems with intermediate chaos.
More broadly, the project team continues its research on quantitative description of intermediate chaos in line with the description of action for the project ICHAOS.