Abstract:
This paper reviews recent results related to rigidity theory for circle
diffeomorphisms with singularities. Both
diffeomorphisms with a break point (sometimes called a
‘fracture-type singularity’ or ‘weak discontinuity’) and critical
circle maps are discussed. In the case of breaks, results are presented on the global
hyperbolicity of the renormalization operator; this property implies the
existence of an attractor of the Smale horseshoe type. It is also shown that
for maps with singularities rigidity is stronger than for
diffeomorphisms, in the sense that rigidity is not violated for
non-generic rotation numbers, which are abnormally well approximable by rationals.
In the case of critical rotations of the circle it is proved that any two such
rotations with the same order of the singular point and the same irrational
rotation number are $C^1$-smoothly conjugate.
Fulltext:
PDF file (389 kB)
