Dynamical systems; Homogeneous spaces; Equidistribution; Effectiveness; Number theory; Dynamics; Entropy; Spectral gap; Effective bounds
Möller M., Pohl A. D. (2013), Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant, in
Ergodic Theory Dynam. Systems, 33(1), 247-283.
Pohl Anke D. (2012), A dynamical approach to Maass cusp forms, in
J. Mod. Dyn., 6(4), 563-596.
Einsiedler Manfred, Mohammadi Amir (2012), A joinings classification and a special case of Raghunathan's conjecture in positive characteristic (with an appendix by Kevin Wortman), in
J. Anal. Math., 116, 299-334.
Kadyrov Shirali (2012), Entropy and escape of mass for Hilbert modular spaces, in
J. Lie Theory, 22(3), 701-722.
Einsiedler Manfred, Kadyrov Shirali (2012), Entropy and escape of mass for SL(3,Z)\SL(3,R), in
Israel J. Math., 190, 253-288.
Kadyrov Shirali (2012), Positive entropy invariant measures on the space of lattices with escape of mass, in
Ergodic Theory Dynam. Systems, 32(1), 141-157.
Einsiedler Manfred, Fishman Lior, Shapira Uri (2011), Diophantine approximations on fractals, in
Geom. Funct. Anal., 21(1), 14-35.
Einsiedler Manfred, Lindenstrauss Elon, Michel Philippe, Venkatesh Akshay (2011), Distribution of periodic torus orbits and Duke's theorem for cubic fields, in
Ann. of Math. (2), 173(2), 815-885.
Polo Fabrizio (2011), Equidistribution of singular measures on nilmanifolds and skew products, in
Ergodic Theory Dynam. Systems, 31(6), 1785-1817.
Einsiedler Manfred, Ward Thomas (2011),
Ergodic theory with a view towards number theory, Springer-Verlag, London 259.
Einsiedler M., Lindenstrauss E. (2010), Diagonal actions on locally homogeneous spaces, Amer. Math. Soc., Providence, 10, 155-241.
Einsiedler Manfred (2010), Effective equidistribution and spectral gap, in
European Congress of Mathematics, ZürichEur. Math. Soc.,, Zurich.
The project concerns equidistribution problems for homogeneous spaces, an area of research linking the theory of Lie groups, algebraic groups, unitary representation, and dynamical systems. Moreover, these problems are closely related to number theory, especially those involving Diophantine solutions to equations and inequalities -- where the solution set and their equations both admit large symmetry groups.The theory of unipotent flows (as developed by Margulis, Ratner, and others) is a very powerful technique to establish equidistribution but has the disadvantage of being non-effective, i.e. there is no error provided for the equidistribution statements. On the other hand representation theoretic or number theoretic arguments are more limited in scope but do provide error rates where they apply. Recently it has become clear that combined arguments can lead to proofs of theorems that are otherwise out of reach. The applicant has obtained in a joint work with Margulis and Venkatesh an effective distribution theorem for closed orbits of semisimple subgroups that currently is still limited to real Lie groups and to semisimple subgroups satisfying a technical assumption (finite centralizer). One part of the project is to generalize this method to the adelic setting and to avoid technical assumptions. This will, in particular, lead to an independent proof of property (tau) for congruence subgroups of semisimple algebraic groups defined over the rational numbers, and to a refinement of a recent joint work of Ellenberg and Venkatesh. Another case of an equidistribution problem is the study of the images of long curves in homogeneous spaces as studied in the recent work of Shah -- it is proposed to study the error rate for instances where the equidistribution has already been established.