Abstract:
This will be an introductory talk.
The centralizer Z(f) of a diffeomorphism f is the set of diffeomorphisms g that commute with f. In other words, Z(f) is the group of symmetries of f, where “symmetries” is meant the classical sense: coordinate changes leave the dynamics of the system unchanged. For certain algebraic examples, they may have exceptional large symmetry group, i.e. the centralizer contains a non-trivial Lie group. For example, the centralizer of the time-1 map f_0 of the geodesic flow of a negatively curved surface contains R, etc.
A natural question is: how about the symmetry group of a perturbation of f_0? This relates to one of the classical questions in perturbation theory: if a diffeomorphism belongs to a smooth flow, which perturbations also belong to a smooth flow. In this talk we will show some background knowledge and basic ideas of some recent results in this direction, joint works with D. Damjanovic and with D. Damjanovic, A. Wilkinson.