On effective equidistribution of ergodic averages for higher step nilflows and higher rank actions

Abstract

In the first part of the thesis, we prove the bounds of ergodic averages for nilflows on general higher step nilmanifolds. It is well known that a flow is not renormalizable on higher step nilpotent Lie algebras and it is not possible to adapt the methods of analysis in moduli space. Instead, we develop the technique called ’scaling method’ for operators behaves like renormalization. On the space of Lie algebra satisfying the "transverse" condition, we obtain the speed of convergence of ergodic averages with the polynomial type of exponent regarding the structure of nilpotent Lie algebras.In the second part, we introduce the results on higher rank actions on Heisenberg nilmanifolds. We develop the method for higher rank action by following the work of A.Bufetov and G.Forni. We construct the finitely-additive measure called Bufetov functional and obtain the prove deviation of ergodic averages on Heisenberg nilmanifolds. As an application, we prove the limit theorem of normalized ergodic integrals.