The essential coexistence phenomenon in Hamiltonian dynamics

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.

Ya. P. was partially supported by NSF grant DMS-1400027. The authors would like to thank Banff International Research Station, where part of the work was done, for their hospitality

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