Abstract:
We develop a Melnikov method for volume-preserving maps with codimension one invariant manifolds. The Melnikov function is shown to be related to the flux of the perturbation through the unperturbed invariant surface. As an example, we compute the Melnikov function for a perturbation of a three-dimensional map that has a heteroclinic connection between a pair of invariant circles. The intersection curves of the manifolds are shown to undergo bifurcations in homology

Abstract:
Explicit formulae are given for the saddle connection of an integrable family of standard maps studied by Y. Suris. When the map is perturbed this connection is destroyed, and we use a discrete version of Melnikov's method to give an explicit formula for the first order approximation of the area of the lobes of the resultant turnstile. These results are compared with computations of the lobe area.

Abstract:
We study quadratic, volume preserving diffeomorphisms whose inverse is also quadratic. Such maps generalize the Henon area preserving map and the family of symplectic quadratic maps studied by Moser. In particular, we investigate a family of quadratic volume preserving maps in three space for which we find a normal form and study invariant sets. We also give an alternative proof of a theorem by Moser classifying quadratic symplectic maps.

Abstract:
We study diffeomorphisms that have one-parameter families of continuous symmetries. For general maps, in contrast to the symplectic case, existence of a symmetry no longer implies existence of an invariant. Conversely, a map with an invariant need not have a symmetry. We show that when a symmetry flow has a global Poincar\'{e} section there are coordinates in which the map takes a reduced, skew-product form, and hence allows for reduction of dimensionality. We show that the reduction of a volume-preserving map again is volume preserving. Finally we sharpen the Noether theorem for symplectic maps. A number of illustrative examples are discussed and the method is compared with traditional reduction techniques.

Abstract:
We use a modification of the parameterization method to study invariant manifolds for difference equations. We establish existence, regularity, smooth dependence on parameters and study several singular limits, even if the difference equations do not define a dynamical system. This method also leads to efficient algorithms that we present with their implementations. The manifolds we consider include not only the classical strong stable and unstable manifolds but also manifolds associated to non-resonant spaces. When the difference equations are the Euler-Lagrange equations of a discrete variational we present sharper results. Note that, if the Legendre condition fails, the Euler-Lagrange equations can not be treated as a dynamical system. If the Legendre condition becomes singular, the dynamical system may be singular while the difference equation remains regular. We present numerical applications to several examples in the physics literature: the Frenkel-Kontorova model with long-range interactions and the Heisenberg model of spin chains with a perturbation. We also present extensions to finite differentiable difference equations.

Abstract:
We study quadratic, volume preserving diffeomorphisms whose inverse is also quadratic. Such maps generalize the H\'enon area preserving map and the family of symplectic quadratic maps studied by Moser. In particular, we investigate a family of quadratic volume preserving maps in three space for which we find a normal form and study invariant sets. We also give an alternative proof of a theorem by Moser classifying quadratic symplectic maps.

Abstract:
Explicit formulae are given for the saddle connection for an integrable family of standard maps studied by Suris. A generalization of Melnikov's method shows that, upon perturbation, this connection is destroyed. We give explicit formula for the first order approximation of the area of the lobes of the resultant turnstile. It is shown that the lobe area is exponentially small in the limit when the Suris map approaches the trivial twist map.

Abstract:
Let $P$ be a set of $n$ points on the plane in general position. We say that a set $\Gamma$ of convex polygons with vertices in $P$ is a convex decomposition of $P$ if: Union of all elements in $\Gamma$ is the convex hull of $P,$ every element in $\Gamma$ is empty, and for any two different elements of $\Gamma$ their interiors are disjoint. A minimal convex decomposition of $P$ is a convex decomposition $\Gamma'$ such that for any two adjacent elements in $\Gamma'$ its union is a non convex polygon. It is known that $P$ always has a minimal convex decomposition with at most $\frac{3n}{2}$ elements. Here we prove that $P$ always has a minimal convex decomposition with at most $\frac{10n}{7}$ elements.