Maps with dense orbits: Ansari's theorem revisited and the infinite torus

The infinite-dimensional torus

Symplectic torus actions with coisotropic principal orbits

In this paper we completely classify symplectic actions of a torus T on a compact connected symplectic manifold (M, σ) when some, hence every, principal orbit is a coisotropic submanifold of (M, σ). That is, we construct an explicit model, defined in terms of certain invariants, of the manifold, the torus action and the symplectic form. The invariants are invariants of the topology of the manif...

Degenerate periodic orbits and homoclinic torus bifurcation.

A one-parameter family of periodic orbits with frequency omega and energy E of an autonomous Hamiltonian system is degenerate when E'(omega) = 0. In this paper, new features of the nonlinear bifurcation near this degeneracy are identified. A new normal form is found where the coefficient of the nonlinear term is determined by the curvature of the energy-frequency map. An important property of t...

Distribution of Periodic Torus Orbits and Duke’s Theorem for Cubic Fields

We study periodic torus orbits on space of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits; for rank 3 lattices, we show that the equivalence classes become uniformly distributed. This is a cubic analogue of Duke’s theorem about the distribution of closed geodesics on the modular surface: suitably interpre...

Semigroups of real functions with dense orbits

Let FI = {f : I → I| f(x) = (Ax + B)/(Cx + D); AD − BC 6= 0}, where I is an interval. For x ∈ I, let Ωx be the orbit of x under the action of the semigroup of functions generated by f, g ∈ FI . Our main result in this paper is to describe all f, g ∈ FI such that Ωx is dense in I for all x.