Modified Schmidt games and a conjecture of Margulis

TL;DR

This paper proves Margulis's conjecture on the abundance of specific exceptional orbits in homogeneous spaces by developing a modified Schmidt game theory, extending classical game frameworks to address complex dynamical systems.

Contribution

It introduces a new variant of Schmidt games tailored for partially hyperbolic flows, enabling the proof of Margulis's conjecture on orbit abundance.

Findings

Proves Margulis's conjecture on exceptional orbit abundance

Develops a modified Schmidt game framework for dynamical systems

Extends classical game theory to new contexts in homogeneous dynamics

Abstract

We prove a conjecture of G.A. Margulis on the abundance of certain exceptional orbits of partially hyperbolic flows on homogeneous spaces by utilizing a theory of modified Schmidt games, which are modifications of $(\alpha,\beta)$-games introduced by W. Schmidt in mid-1960s.