# On the total disconnectedness of the quotient Aubry set

## TL;DR

This paper proves that for specific Lagrangians, the quotient Aubry set is totally disconnected, and explores its connection to a Morse-Sard type property for Hamilton-Jacobi critical subsolutions.

## Contribution

It establishes the total disconnectedness of the quotient Aubry set for certain Lagrangians and links this to a Morse-Sard type property in Hamilton-Jacobi theory.

## Key findings

- Quotient Aubry set is totally disconnected for certain Lagrangians
- Connection between Aubry set disconnectedness and Morse-Sard property
- Insights into the structure of critical subsolutions of Hamilton-Jacobi equations

## Abstract

In this paper we show that the quotient Aubry set associated to certain Lagrangians is totally disconnected (i.e., every connected component consists of a single point). Moreover, we discuss the relation between this problem and a Morse-Sard type property for (difference of) critical subsolutions of Hamilton-Jacobi equations.