哈密顿系统的遍历性

Certain systems of three falling balls satisfy the Chernov-Sinai ansatz

The system of falling balls is an autonomous Hamiltonian system with a smooth invariant measure and non-zero Lyapunov exponents almost everywhere. Since almost three decades, the question of ergodicity is still open. The subject of this work is to contribute to the solution of the ergodicity conjecture for three falling balls with a specific mass ratio. The latter is executed in the following three points: First, we prove the Chernov-Sinai ansatz. Second, we prove that there is an abundance of least expanding points and, third, we explain that the proper alignment condition can still be verifi…

Piecewise Linear Hamiltonian Flows Associated to Zero-Sum Games: Transition Combinatorics and Questions on Ergodicity

In this paper we consider a class of piecewise affine Hamiltonian vector fields whose orbits are piecewise straight lines. We give a first classification result of such systems and show that the orbit-structure of the flow of such a differential equation is surprisingly rich.

Whispering gallery orbits in Sinai oscillator trap

Experimental realizations of trapping Bose Einstein condensate lead to a Hamiltonian system of a classical particle bouncing off a convex scatterer in the field of an attracting potential. It is shown by application of KAM theory that under some natural conditions there exists positive measure of quasiperiodic solutions near the boundary.

一般势场中的低能动力学：双曲周期轨道与非遍历性

Low-energy dynamics in generic potential fields: Hyperbolic periodic orbits and non-ergodicity

本文证明在二维环面上一般光滑势的自然哈密顿系统低能级中，双曲周期轨道与不变环面共存，揭示低能动力学的非遍历性。

A Remark on Unique Ergodicity and the Contact Type Condition

We prove that for a broad class of exact symplectic manifolds including ${\mathbb R}^{2m}$ the Hamiltonian flow on a regular compact energy level of an autonomous Hamiltonian cannot be uniquely ergodic. This is a consequence of the Weinstein conjecture and an observation that a Hamiltonian structure with non-vanishing self-linking number must have contact type. We apply these results to show that certain types of exact twisted geodesic flows cannot be uniquely ergodic.

Liouville ergodicity of linear multi-particle hamiltonian system with one marked particle velocity flips

We consider multi-particle systems with linear deterministic hamiltonian dynamics. Besides Liouville measure it has continuum of invariant tori and thus continuum of invariant measures. But if one specified particle is subjected to a simple linear deterministic transformation (velocity flip) in random time moments, we prove convergence to Liouville measure for any initial state. For the proof it appeared necessary to study non-linear transformations on the energy surface.