Notes on $s$ and $u$-states for cocycles over partially hyperbolic maps

TL;DR

This paper proves that the sets of $s$ and $u$-states for cocycles over partially hyperbolic maps are closed within the space of invariant measures, providing insight into their stability.

Contribution

It establishes the closure property of $s$ and $u$-states for cocycles over partially hyperbolic systems, a result not previously known.

Findings

$s$ and $u$-states form closed sets in the space of invariant measures

The result applies to cocycles over partially hyperbolic maps

Provides foundational understanding of invariant measure stability

Abstract

In these notes we prove that the $s$ or $u$-states of cocycles over partially hyperbolic maps are closed in the space of invariant measures.