Non-uniform ergodic properties of Hamiltonian flows with impacts. (English) Zbl 1525.37027
In this paper the ergodic properties of two uncoupled oscillators are studied. The motion of the associated particle is thereby restricted by star-shaped polygons with only horizontal or vertical boundaries. When the particle hits the boundary it is reflected according to the law of reflection. The iso-energy level sets of this systems change non-trivially. In particular it is proven that for some partial energies the corresponding translation surfaces posses a genus strictly greater than 1. Throughout the paper various configurations of the two uncoupled oscillators are discussed.
The paper is well written and introduces all necessary concepts in a highly illustrative way. In the reference list one finds many well chosen articles supplementing this work.
The paper is well written and introduces all necessary concepts in a highly illustrative way. In the reference list one finds many well chosen articles supplementing this work.
Reviewer: Daniel Jaud (München)
MSC:
| 37C83 | Dynamical systems with singularities (billiards, etc.) |
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 37E35 | Flows on surfaces |
| 37J06 | General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants |
| 37J20 | Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
Hamiltonian impact systems; quasi-integrable systems; unique ergodicity; translation surfaces; piecewise smooth dynamicsReferences:
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