# The essential coexistence phenomenon in Hamiltonian dynamics

## TL;DR

This paper constructs a Hamiltonian flow on a 4D manifold demonstrating the coexistence of regular and chaotic dynamics on an energy surface, with chaotic behavior in an open dense set and regular behavior on the boundary.

## Contribution

It provides a novel example of Hamiltonian dynamics exhibiting essential coexistence of regular and chaotic behaviors on the same energy surface.

## Key findings

- Existence of an open dense invariant set with Bernoulli flow and positive Lyapunov exponents.
- Boundary of the invariant set has zero Lyapunov exponents.
- Demonstrates complex coexistence phenomena in Hamiltonian systems.

## Abstract

We construct an example of a Hamiltonian flow $f^t$ on a $4$-dimensional smooth manifold $\mathcal{M}$ which after being restricted to an energy surface $\mathcal{M}_e$ demonstrates essential coexistence of regular and chaotic dynamics that is there is an open and dense $f^t$-invariant subset $U\subset\mathcal{M}_e$ such that the restriction $f^t|U$ has non-zero Lyapunov exponents in all directions (except the direction of the flow) and is a Bernoulli flow while on the boundary $\partial U$, which has positive volume all Lyapunov exponents of the system are zero.