Surfaces of sections are a classical tool in the study of 3-dimensional dynamical systems. Their use goes back to the work of Poincar´ and Birkhoﬀ. e In the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of ﬁnitely many distinguished periodic solutions. Such a system is established for nondegenerate Reeb ﬂows on the tight 3-sphere by means of pseudoholomorphic curves.
We prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level. Since typically there is dynamical localization at the edges of each disordered-broadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, namely a boundary point between the localization and delocalization regimes, which we prove to converge to the corresponding Landau level as either the magnetic ﬁeld goes to inﬁnity or the disorder goes to zero. ...
One of the important tools of geometric mechanics is reduction theory (either
Lagrangian or Hamiltonian),which provides a well-developed method for dealing
with dynamic constraints. In this theory the dynamic constraints and the sym-
metry group are used to lower the dimension of the system by constructing an
associated reduced system. We develop the Lagrangian version of this theory for
nonholonomic systems in this paper. We have focussed on Lagrangian systems
because this is a convenient context for applications to control theory. ...