Moscow Mathematical Journal

Volume 12, Issue 4, October–December 2012 pp. 705–717.

On Products of Skew Rotations

Authors: M.D. Arnold (1, 2), E.I. Dinaburg (1, 3), G.B. Dobrushina (1), S.A. Pirogov (1), and A.N. Rybko (1)
Author institution: (1) Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Bolshoi Karetny per. 19, Moscow, 127994, Russia
(2) International Institute of Earthquake Prediction Theory and Mathematical Geophysics of the Russian Academy of Sciences, Profsoyuznaya str., 84/32, Moscow, 117997, Russia
(3) Schmidt Institute of Physics of the Earth of the Russian Academy of Sciences, B. Gruzinskaya str., 10, Moscow, 123995, Russia
Summary: Let $\{S_1^t\},\dots,\{S_n^t\}$ be the one-parametric groups of shifts along the orbits of Hamiltonian systems generated by time-independent Hamiltonians $H_1,\dots, H_n$ with one degree of freedom. In some problems of population genetics there appear planar transformations having the form $S^{h_n}_n\cdots S_1^{h_1}$ under some conditions on Hamiltonians $H_1,\dots,H_n$. In this paper we study asymptotical properties of trajectories of such transformations. We show that under classical non-degeneracy condition on the Hamiltonians the trajectories stay in the invariant annuli for generic combinations of lengths $h_1,\dots, h_n$, while for the special case $h_1+\dots+h_n=0$ there exists a trajectory escaping to infinity.

2010 Mathematics Subject Classification. 37J40, 37J15, 37M05
Contents Full-Text PDF