Published: Feb. 14, 2013

Robust Invariant Tori, Multidimensional Farey Trees, and Algebraic Irrationals

Invariant tori are prominent features of Hamiltonian and symplectic dynamical systems that are integrable or nearly so. Their prominence arises in part because of the celebrated KAM theorem which shows the structural stability of certain “very irrational” (Diophantine) tori for “smooth enough” systems that are “anharmonic” (satisfy a twist condition). This robustness is responsible for the long time correlations and slow transport in chaotic Hamiltonian dynamics.

Each preserved torus is associated with a rotation (frequency ratio) vector that characterizes the dynamics on the torus. When the rotation vector is a scalar (two-tori for flows, one-tori for maps), the torus can be approximated in an optimal way by a sequence of periodic orbits obtained through the continue fraction expansion. Properties of this expansion, such as it being of “bounded type” are closely related to the robustness of the associated torus. In particular certain quadratic irrational numbers, the “nobles” are conjectured to be locally, maximally robust.

Generalization of these ideas to rotation vectors has been difficult for a number of reasons. One is that there is no satisfactory generalization of the continued fraction--indeed classical number theory shows that certain reasonable requirements (periodicity and giving best approximants) are incompatible. Nevertheless it has been long been conjectured that vectors in certain algebraic fields (e.g. cubic fields for the two-dimensional case) will replace the nobles in higher dimensions. We will discuss some generalizations of the continued fraction and a related idea, the Farey tree, to higher dimensions. Generalized Farey trees such as the version of Ostlund and Kim, abandon the best approximant property in favor of periodicity and have some nice properties for certain algebraic irrationals.

Jim Meiss

Applied MathematicsUniversity of Colorado Boulder

Date and time: 

Thursday, February 14, 2013 - 3:15pm