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.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2019
- DOI:
- 10.48550/arXiv.1901.07713
- arXiv:
- arXiv:1901.07713
- Bibcode:
- 2019arXiv190107713C
- Keywords:
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- Mathematics - Dynamical Systems;
- 37D35;
- 37C45;
- 37C40;
- 37D20
- E-Print:
- 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