Equivariant Kolmogorov-Arnold-Moser theory

Abstract

KAM theory is a collection of theorems and approaches to describe the stability of certain invariant tori within nearly integrable Hamiltonian systems. The invariant tori are associated with a set of torus frequencies, and a key result in the theory is that these frequencies must satisfy a Diophantine condition, with no rational resonances amongst the frequencies. In this paper we investigate the additional property of equivariance with a discrete symmetry group, Γ. The setting then becomes Γ-Equivariant, nearly integrable, Hamiltonian, ODEs. In this setting the Diophantine condition is no longer valid as the symmetry constraints imposed by Γ, generically force the frequencies to be repeated. Utilizing equivariant-singularity and linear representation theory we develop an alternative Γ-Diophantine condition and show that by modifying the Diophantine assumption in this way KAM theory can be applied to this new setting. Moreover, we demonstrate that this condition can be utilized in the Hyperbolic, Γ- Equivariant case.