Digits, distributions, and rigidity in dynamical systems

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Dynamical systems tools and techniques are indispensable to modern science and technology as they enable accurate modelling of processes ranging from space flight to cell growth to economic shocks. The wide scope of dynamical systems and the vast amounts of data generated by them have made the study of their quantitative and computational aspects an important research area. To this area the proposed program will contribute in several ways. The research topics can roughly be grouped as they relate, respectively, to digits, distributions, and rigidity.***Digits. The program's main long-term goal is to establish a comprehensive theory of digits, their properties and distributions, for dynamical systems. A prominent phenomenon in this context is the surprising ubiquity of one particular logarithmic distribution often referred to as Benford's Law. Regarding the latter, many mathematical questions remain which the program will address, building not least on results obtained under the applicant's previous Discovery Grant. Recent advances in the theory of dynamical systems will be utilized to precisely quantify the randomness inherent to the digits generated by familiar one- and multi-dimensional systems.***Distributions. Another important goal is to carry out an in-depth analysis of probabilities induced by digit statistics, notably on the circle, the discretization and (constrained) approximation properties of which are only partly understood at present. This analysis will include the study of novel quantitative concepts such as distortion (with respect to scale, base, or sum) and spread of probability measures for which no theoretical framework exists at present, but also refinements of classical results in uniform distribution theory.***Rigidity. A third common theme of the program is rigidity. Informally put, in case of rigidity far fewer objects with certain properties exist than might reasonably be expected. Many deep rigidity results have been established in dynamics. To this existing zoo, the program will introduce two new species, namely digit rigidity and equivalence rigidity. Both forms of rigidity provide unexpected insights into the behavior of linear systems, arguably the simplest class of dynamical systems widely used throughout the sciences.***The core parts of the proposed program are highly interrelated. Together, they provide a deepened understanding of important quantitative aspects of dynamical systems. The results established under this program will be of interest to other areas of pure and applied mathematics, notably number theory, optimal transport, and statistics. They will lead to new or more reliable statistical tests and data plausibility checks (e.g., with regard to potential fraud in accounting, election, or scientific data) as well as to improved numerical schemes that will benefit all scientists who use dynamical systems as models for real-world processes.*** *** *** *** *** *** *** **