Simple and complex dynamics for circle maps (Q1325601)

The aim of the paper is to extend the characterisation of the complex and simple interval maps to circle maps. It is well known that each interval map satisfies one and only one of the following two conditions: the map is horseshoe; or the chain recurrent set of the map is a union of all simple sets of map. The main result on the paper is the following: Let \(f\) be a circle map. Then \(f\) satisfies one and only one of the following three conditions: (a) \(f\) is horseshoe; (b) There exist \(n > 0\) such that the omega-limit set of \(f^ n\) is a simple set for each point form the circle; (c) The set of periods of all periodic points of \(f\) is empty. -- An interesting geometric view-point of condition (b) is represented in section 4 of this paper. The authors use standard degree-technique to prove main results in the paper.