Dynamical complexity and symplectic integrability

TL;DR

This paper introduces new numerical invariants called complexity indices for integrable Hamiltonian systems, providing tools to measure their polynomial growth rates and classifying a new class of systems called decomposable systems.

Contribution

It defines complexity and weak complexity indices, introduces decomposable systems, and computes these indices for specific Hamiltonian and gradient systems.

Findings

Weak complexity index is smaller than the degrees of freedom for decomposable systems.

Explicit calculations of complexity indices for Morse Hamiltonian systems on surfaces.

Hamiltonian systems with non-degenerate integrals are examples of decomposable systems.

Abstract

We introduce two numerical conjugacy invariants for dynamical systems -- the complexity and weak complexity indices -- which are well-suited for the study of "completely integrable" Hamiltonian systems. These invariants can be seen as "slow entropies", they describe the polynomial growth rate of the number of balls (for the usual "dynamical" distances) of coverings of the ambient space. We then define a new class of integrable systems, which we call decomposable systems, for which one can prove that the weak complexity index is smaller than the number of degrees of freedom. Hamiltonian systems integrable by means of non-degenerate integrals (in Eliasson-Williamson sense), subjected to natural additional assumptions, are the main examples of decomposable systems. We finally give explicit examples of computation of the complexity index, for Morse Hamiltonian systems on surfaces and for two-dimensional gradient systems.