Integrable mappings.

*(Russian)*Zbl 0746.58033The paper deals with the problems of integrable systems of differential equations and mappings. The different conceptions of integrable mapping are examined. In the first chapter the Lagrangian systems with discrete time are considered. Specifically results connected with the Liouville theorem are described. For integrable systems of classical mechanics the paper gives results which are received by J. Moser and the author. There are the following examples: 1) the Neumann problem for the movement of the point on the sphere with quadratic potential, 2) the problem of billiards inside an ellipsoid, 3) the discrete analogon of the problem of rigid body movement.

In the second chapter the dynamics of integrable mappings is examined. Grave attention is given to polynomial mappings \(f: \mathbb{C}^ n\to \mathbb{C}^ n\). A polynomial \(f\) \((n=1)\) is called integrable if there exists a polynomial \(g\) such that \(fg=gf\) and the trajectories of \(f\) and \(g\) are not intersected. The result which describes the class of all integrable polynomials is given. Here the possible definitions of integrable polynomial mappings are given for the case \(n>1\). Examples exist which exhibit a difference between dynamics of polynomial and rational mappings. Applications follow.

In the second chapter the dynamics of integrable mappings is examined. Grave attention is given to polynomial mappings \(f: \mathbb{C}^ n\to \mathbb{C}^ n\). A polynomial \(f\) \((n=1)\) is called integrable if there exists a polynomial \(g\) such that \(fg=gf\) and the trajectories of \(f\) and \(g\) are not intersected. The result which describes the class of all integrable polynomials is given. Here the possible definitions of integrable polynomial mappings are given for the case \(n>1\). Examples exist which exhibit a difference between dynamics of polynomial and rational mappings. Applications follow.

Reviewer: G.Osipenko (St.Petersburg)

##### MSC:

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

54H20 | Topological dynamics (MSC2010) |