Ergodic theory on homogeneous flows and Ratners theorems

Author(s)

Thiago Rodrigo Ramos

• Document type: Master's Dissertation
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2018-06-14

Examining board members

Daniel Smania Brandão; Carlos Alberto Maquera Apaza; Lino Anderson da Silva Grama; Eduardo Tengan

• Advisor: Daniel Smania Brandão

Abstract

In this work, we prove a particular case of the Ratners measure classification theorem, which tell us that if X = Γ\\G is an homogeneous space, where G is a Lie group and Γ is a lattice of G, then given any unipotent group U of G, we can classify the measures that are ergodic with respect to the translation group action of U in X In addition to the Ratners measure classification theorem, we talk about the Ratners equidistribuition theorem and the Ratners orbit closure theorem, which tell us how the orbit due the action by translation by the group U are and how the dynamics in X is, in an Ergodic Theory point of view. While we didnt prove the last two Ratners theorems, we exhibit an important application of the Ratners orbit closure theorem in number theory, proving the Oppeinheim Conjecture, also know as Margullis Theorem. (AU)

• FAPESP's process: 16/01411-2 - Ergodic theory of homogeneous flows
• Grantee: Thiago Rodrigo Ramos
• Support Opportunities: Scholarships in Brazil - Master