We construct an example of a Hamiltonian flow $f^t$ on a four-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 for 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.

中文翻译：

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哈密​​顿动力学中的本质共存现象

我们构建了一个哈密顿流的例子$f^t$在四维光滑流形上$\数学{M}$在被限制在一个能量表面之后$\mathcal {M}_e$展示了规律和混沌动力学的本质共存，即存在一个开放和密集的$f^t$- 不变子集$U\subset \mathcal {M}_e$这样的限制$f^t|U$在所有方向（除了流动方向）都有非零李雅普诺夫指数，并且是伯努利流，而在边界上$\部分U$，具有正体积，系统的所有 Lyapunov 指数都为零。