Positive-Entropy Integrable Systems and the Toda Lattice, II

TL;DR

This paper constructs integrable convex Hamiltonian systems on specific torus bundles, utilizing Lax representations of Toda lattices, and classifies these systems based on their entropy and automorphism groups.

Contribution

It introduces new integrable Hamiltonian systems on torus bundles and links their classification to automorphism groups and entropy.

Findings

Constructed integrable convex Hamiltonians on torus bundles.

Connected system classification to abelian groups of Anosov automorphisms.

Used Lax representation of Toda lattices for system analysis.

Abstract

This note constructs completely integrable convex Hamiltonians on the cotangent bundle of certain k-dimensional torus bundles over an l-dimensional torus. A central role is played by the Lax representation of a Bogoyavlenskij-Toda lattice. The classification of these systems, up to iso-energetic topological conjugacy, is related to the classification of abelian groups of Anosov toral automorphisms by their topological entropy function.