FWF - SYMDYNAR - Symbolic dynamics and arithmetic expansions

Wider research context / theoretical framework. In France as well as in Austria, research on dynamical systems with special emphasis on arithmetic expansions like numeration systems and continued fractions has a long tradition. We will structure the project into four tasks: 1) Infinite interval exchange transformations, 2) Dynamics of continued fractions, 3) Arithmetical and spectral properties of shift spaces, 4) Normal numbers and arithmetic functions. The relevance of the directions of research in these tasks is testified by major breakthroughs that have been achieved in recent years. For instance the work on interval exchange transformations by Avila and his coworkers, the progress on the Littlewood conjecture by Einsiedler, Katok, and Lindenstrauss or the proof of an old conjecture of Gelfond by Mauduit and Rivat with implications to the famous Sarnak conjecture has strong relations to the number theoretical part of our project. Substitutions and aperiodic tilings, which forms another important part of our project, found a surprising application in material science in 1982, when Dan Shechtman (Nobel Laureate of 2011) discovered particular alloys with icosahedral symmetry in their diffraction spectrum, contrary to what is possible in crystalline (i.e., periodic) materials. Approach/methods. We will use symbolic codings to understand smooth dynamical systems like interval exchange maps, toral translations, toral automorphisms (stationary setting) and Anosov families (non stationary setting), geodesic flows on homogeneous spaces and Lie Groups in general. Representations of these systems in terms of symbolic systems (like sofic shifts or shifts of finite type) are very useful to gain properties (eigenvalues, invariants or recurrence) of the original systems and in our case have strong relations to numeration and continued fractions. Such codings can be found e.g. by using partitions (e.g. Markov ones) or Poincaré sections. Our codings admit scale invariance that allows modeling them by renormalization schemes. As renormalization dynamics is hyperbolic it can be understood by applying the well developed theory of hyperbolic dynamics. We are thus able to associate cocycles with our systems and study their Lyapunov exponents. Applications include arithmetics, normality, computability, cryptography, and aspects of mathematical physics. Level of originality / innovation. Our aim is to strengthen the important bridge between number theory and dynamics which will lead to new insights in the theory of continued fractions and Diophantine approximation.