Abstract:

The so-called "critical circle maps" are orientation-preserving C^3

circle homeomorphisms having a non-flat critical point (they belong to

the boundary of the C^3 diffeomorphisms).

The "Rigidity Conjecture" for critical circle maps with irrational

rotation number was formulated in the early eighties after several

works of Feigenbaum, Kadanoff, Lanford, Rand and Shenker among others,

and it was proved to be true in the real-analytic category by de

Faria-de Melo 2000, Yampolsky 2003 and Khanin-Teplinsky 2007.

Recently, on a joint work with Welington de Melo, we proved the

rigidity conjecture for C^3 critical circle maps with irrational

rotation number of bounded type (arXiv:1303.3470).

More recently, we were able to get rid of the bounded combinatorics

condition, thus extending the rigidity to any irrational rotation

number: inside each topological class of C^3 critical circle maps, the

degree of the critical point is the unique invariant of the smooth

conjugacy classes. Work in progress with Marco Martens and Welington

de Melo.