Let S 1 be a circle of perimeter equal to 1 (not radius 1) .Let...
Let be a circle of perimeter equal to 1 (not radius 1) .Let be an arbitrary homeomorphism. By uniform continuity, it always possible to find an such that implies
where d denote the shortest distance between two points on the circle.
My question is: Is it always possible to find an such that for any arc (or call it interval if you like) on such that implies that
In other words, do homeomorphisms always send very short arcs to short arcs, rather than long arcs?
Since a homeomorphism can be expanding or contracting, I am really puzzled by this. There must be some tricks I don't know.