Flow-invariant sets and critical point theory

On invariant sets topology

In this paper, we introduce and study a new topology related to a self mapping on a nonempty set.Let X be a nonempty set and let f be a self mapping on X. Then the set of all invariant subsets ofX related to f, i.e. f := fA X : f(A) Ag P(X) is a topology on X. Among other things,we nd the smallest open sets contains a point x 2 X. Moreover, we find the relations between fand To f . For insta...

Fixed Point Indices and Invariant Periodic Sets of Holomorphic Systems

This note presents a method to study center families of periodic orbits of complex holomorphic differential equations near singularities, based on some iteration properties of fixed point indices. As an application of this method, we will prove Needham’s theorem in a more general version. 1. Fixed point indices of holomorphic mappings Let C be the complex vector space of dimension n, let U be a...

Similarity and Affine Invariant Distances Between 2D Point Sets

Order type invariant labeling and comparison of point sets

We consider the problem of computing an order type invariant labeling for a given set of n points. In 2D, such a labeling can be constructed in O(hn2) time, where h is the size of the smallest convex layer. In 3D the time complexity is O(n3 log n) if the point set is in general position1. This is useful to test if two point sets have the same order type within the same time bounds. It can also ...

Inequalities of Critical Point Theory

A purpose of critical point theory is the counting of critical points of functions. Principal theorems in the subject state in precise terms that topological complexity of the underlying space is reflected in the existence and nature of critical points of any smooth real-valued function defined on the space. The initial development of critical point theory is peculiarly the work of one man, Mar...