# Central Limit theorem and cohomological equation on homogeneous spaces

## TL;DR

This paper extends Livšic type results and proves a central limit theorem for Birkhoff averages in partially hyperbolic homogeneous space systems, revealing nonzero variance under certain conditions.

## Contribution

It introduces a Livšic type theorem for noncompact, nonaccessible homogeneous systems and establishes a central limit theorem for horospherical orbit averages.

## Key findings

- Livšic type result extended to noncompact systems
- Central limit theorem proven for Birkhoff averages
- Variance is nonzero for functions with nonzero mean

## Abstract

The dynamics of one parameter diagonal group actions on finite volume homogeneous spaces has a partially hyperbolic feature. In this paper we extend the Liv\v{s}ic type result to these possibly noncompact and nonaccessible systems. We also prove a central limit theorem for the Birkhoff averages of points on a horospherical orbit. The Liv\v{s}ic type result allows us to show that the variance of the central limit theorem is nonzero provided that the test function has nonzero mean with respect to an invariant probability measure.