Renormalization horseshoe and rigidity for circle diffeomorphisms with breaks
The authors present an almost complete renormalization theory for circle homeomorphisms with breaks, this is, homeomorphisms which are smooth everywhere except at a single point where the first derivative has a jump discontinuity. After presenting the basic renormalization construction for orientation-preserving circle homeomorphisms, the authors define circle diffeomorphisms with breaks and describe the properties of their renormalizations. They prove that the corresponding two-dimensional transformation (a linear-fractional family) has strong hyperbolic properties which allows them to construct the full renormalization horseshoe. Furthermore, a complete description of the uniformly hyperbolic horseshoe in the two-dimensional invariant manifold consisting of linear-fractional commuting pairs, to which the sequences of renormalizations converges, is given. In the last section of this paper, the authors show the smoothness of the conjugacy of two circle diffeomorphisms with breaks with the same size of break and the same irrational rotation number of a half-bounded type.
scientific article; zbMATH DE number 6176794