Homeomorphisms of the circle were first considered by Poincare* who used them to obtain qualitative results for a class of differential equations on the torus. He classified those which have a dense orbit by showing that they are topologically equivalent to a rotation through an angle incommensurable with IT. However, Denjoy showed that there exist homeomorphisms of the circle without periodic ...

Rotation Theory has its roots in the theory of rotation numbers for circle homeomorphisms, developed by Poincaré. It is particularly useful for the study and classification of periodic orbits of dynamical systems. It deals with ergodic averages and their limits, not only for almost all points, like in Ergodic Theory, but for all points. We present the general ideas of Rotation Theory and its ap...

1 Overview, review and first definitions 3 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Calculus and linear algebra review . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Distances in R and straight segments . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

We complete the program begun by Brin and Squier of characterising conjugacy in Thompson’s group F using the standard action of F as a group of piecewise linear homeomorphisms of the unit interval.

In this article, we introduce the concept of co-covering which is the dual of covering concept. Then, we prove several theorems being similar to the theorems that have been de-veloped for the covering concept. For example, we provide the lifting criterion for co-coverings of a topological space X, which helps us to classify them by subgroups of the group of all homeomorphisms of X.

We develop a local theory of extremal analytic discs attached to a strictly pseudoconvex manifold of higher codimension in complex space. We then apply the results to prove the regularity of CR homeomorphisms of such manifolds.

We show that there are homeomorphisms of closed oriented genus g surfaces Σg which are fiber-preserving with respect to an irregular branched covering Σg → S2 and isotopic to the identity, but which are not fiber-isotopic to the identity.