Title: Ergodic and mixing properties of infinite measure preserving transformations arising from tied down distributional limits.
Abstract: For certain infinite ergodic transformations, the absolutely normalized ergodic sums (occupation times) may converge to a non-trivial distributional limit as in e.g. the Darling-Kac theorem. I’ll discuss some ergodic and mixing properties arising from tied down convergence (i.e. conditioned to times of occupation). Joint work with Toru Sera (arXiv:1910.09846).
Title: Rotated odometers
Abstract: We consider a class infinite interval exchange transformations (IETs) obtained as a composition of a finite IET and the von Neumann-Kakutani map, and study their dynamical and ergodic properties by means of the assoicated Bratteli-Vershik diagram. In particular, we study the question whether such a system or its minimal subsystem factor onto the dyadic odometer. This is joint work with Henk Bruin.
Title: Effective equidistribution of orbits of semisimple groups on congruence quotients
Abstract: In this short talk, we discuss polynomially effective versions of a theorem of Mozes and Shah from the 90’s describing limits of measures on homogeneous spaces that are invariant and ergodic under unipotent flows. Remarkable work of Einsiedler, Margulis and Venkatesh from ’09 shows that orbits of a semisimple group on congruence quotients are equidistributed with a polynomial error rate in the volume assuming that the centralizer of the acting group is trivial. While this centralizer assumption does not appear to be essential to the method, it has proven to be challenging to remove. Building on effective avoidance principles recently developed by Lindenstrauss, Margulis, Mohammadi and Shah, we can remove the centralizer assumption under a Diophantine condition.
E. Arthur Robinson
Title: Continued fraction normality for the Minkowski question mark function
Abstract: In 1981, Adler, Kean and Smorodinski constructed an explicit number that is continued fraction normal. This number is analogous to the 1933 Champernowne normal number base 10. In particular, it is a generic point for the continued fraction map G with the (absolutely continuous) Gauss measure g. But besides the Gauss measure g, another ergodic invariant measure for the continued fraction map G is the measure q whose distribution is the famous Minkowski “question mark” function Q. This function is continuous and strictly increasing, but satisfies Q'(x)=0 a.e.. We will display an explicit Champernowne style normal number for this measure. The proof uses the normality of the binary Champernowne number and the Kepler tree on the rationals. This is joint work with Karma Dajani and Mathijs de Lepper of Utrecht University.
Title: Transversal to horocycle flow on the moduli space of doubled slit tori
Abstract: In this short talk we introduce an explicit transversal to horocycle flow on the moduli space of doubled slit tori and hint at some applications This talk does not assume any prior knowledge of the words in the title.
Title: Classification of topological systems which split into uniquely ergodic subsystems
Abstract: We study connections between several similarly looking properties of (invertible)
topological dynamical systems \((X,T)\):
(a) Each point is generic for an ergodic measure;
(b) The system splits into uniquely ergodic subsystems;
(c) The measure center of the system splits into strictly ergodic subsystems;
(d) The Cèsaro means of continuous functions converge uniformly.
We also have a Jewett-Krieger type theorem for non-ergodic measure-preserving systems, but I will only have time to state the result without a proof.