On Periodic Maps which Respect A Symplectic Structure

Periodic Maps Which Preserve a Complex Structure

On the Structure of Periodic Arithmetical Maps

This is an introduction to the algebraic theory of periodic arithmetical maps. The topic is connected with number theory, combinatorics, algebra and analysis. 1. Various Backgrounds I. Combinatorial Background. For n ∈ Z = {1, 2, 3, · · · } and a ∈ R(n) = {0, 1, · · · , n− 1}, let a(n) = a+ nZ = {x ∈ Z : x ≡ a (mod n)} = {· · · , a− n, a, a+ n, · · · } and call it a residue class with modulus n...

Circle valued momentum maps for symplectic periodic flows

We give a detailed proof of the well-known classical fact that every symplectic circle action on a compact manifold admits a circle valued momentum map relative to some symplectic form. This momentum map is Morse-Bott-Novikov and each connected component of the fixed point set has even index. These proofs do not appear to be written elsewhere.

Relative periodic points of symplectic maps: persistence and bifurcations

In this paper we study symplectic maps with a continuous symmetry group arising by periodic forcing of symmetric Hamiltonian systems. By Noether’s Theorem, for each continuous symmetry the symplectic map has a conserved momentum. We study the persistence of relative periodic points of the symplectic map when momentum is varied and also treat subharmonic persistence and relative subharmonic bifu...

Symplectic maps

Here Dfij = ∂fi/∂zj is the Jacobian matrix of f , J is the Poisson matrix, and I is the n × n identity matrix. Equivalently, Stokes’ theorem can be used to show that the loop action, A[γ] = ∮ γ pdq, is preserved by f for any contractible loop γ on X. If f preserves the loop action for all loops, even those that are not contractible, then it is exact symplectic. When n = 1, the symplectic condit...