Nekhoroshev theory and discrete averaging

TL;DR

This paper proves Nekhoroshev's theorem for quasi-integrable symplectic maps using a novel discrete averaging approach, avoiding normal form transformations, and demonstrates how near-identity maps embed into Hamiltonian flows with exponentially small errors.

Contribution

It introduces a new proof of Nekhoroshev's theorem for symplectic maps based on discrete averaging, differing from classical normal form methods.

Findings

Proof of Nekhoroshev theorem for symplectic maps

Embedding of near-identity maps into Hamiltonian flows with small error

Use of discrete averaging instead of normal form transformations

Abstract

This paper contains a proof of the Nekhoroshev theorem for quasi-integrable symplectic maps. In contrast to the classical methods, our proof is based on the discrete averaging method and does not rely on transformations to normal forms. At the centre of our arguments lies the theorem on embedding of a near-the-identity symplectic map into an autonomous Hamiltonian flow with exponentially small error.