Rigidity of critical circle maps.
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