Conservative maps: Reversibility, invariants and approximation

Conservative dynamics refers to dynamical systems that are measure-preserving. A special class of such systems are hamiltonian flows and their discrete analogue, symplectic maps. Reversibility as well as integrability are symmetry-like properties that, although not necessarily limited to the conservative case, attain special relevance in this context.;Polynomial automorphisms can be counted among the simplest instances of non-linear maps; yet, as dynamical systems, they may exhibit the full range of complexity from integrability to chaos. This trait makes polynomial automorphisms especially suited to approximate the dynamics of more general systems. In particular we discuss approximation by polynomial shears and by products of Lie transforms. We study reversibility in the group of polynomial automorphisms of the plane and show that in this group reversors always have finite order. We use Jung's theorem and the generalized Henon maps introduced by Friedland and Milnor [19] to provide normal forms for reversible polynomial automorphisms. We show that every such map is conjugate to a map of the form &parl0;h-11&cdots;h-1 m&parr0;r1&parl0;hm&cdots;h1 &parr0;r0, where each hi is a generalized Henon transformation and r0, r 1 are reversors of a particularly simple form. Some of the dynamical consequences of reversibility are also considered.;Following techniques due to Suris [55] we construct a family of rational diffeomorphisms of R3 that are volume and orientation-preserving and possess a polynomial invariant. Although when restricted to an invariant surface these maps behave locally as area-preserving maps, their dynamics is not equivalent to that of a plane diffeomorphism. We also obtain families of volume-preserving and orientation-reversing diffeomorphisms with an invariant; unlike the orientation-preserving case some of these diffeomorphisms turn out to be polynomial.