On the Hausdorff Dimension of the Mather Quotient

TL;DR

This paper investigates the Hausdorff dimension of the Mather quotient under certain conditions, demonstrating its small size and exploring implications in dynamical systems.

Contribution

It establishes bounds on the Hausdorff dimension of the Mather quotient and applies these results to problems in dynamics.

Findings

Mather quotient has small Hausdorff dimension under specific assumptions

Results have implications for the understanding of dynamical systems

Provides new bounds linking geometry and dynamics

Abstract

Under appropriate assumptions on the dimension of the ambient manifold and the regularity of the Hamiltonian, we show that the Mather quotient is small in term of Hausdorff dimension. Then, we present applications in dynamics.