Speaker
Description
Rigidity is a central question investigated in one dimensional dynamics: we say that a class of one dimensional maps is rigid when a topological conjugacy between two of them has automatically further regularity properties. In this talk we want to highlight how a notion of 'combinatorial rotation number' borrowed from the study of Interval exchange transformations (IETs) can help investigate rigidity in different settings, by discussing two recent results for multi-critical circle maps and affine interval exchange maps (AIETs) respectively.
For multi-critical circle maps, assuming exponential convergence of renormalization, we show that two maps with the same signature, under a full measure condition of the latter, are C^(1+a) conjugated (joint work with Estevez and Trujillo). In the setting of AIETs, we describe a class of AIETs which are C^10 but not C^1 conjugated (joint work with Trujillo). In both works, a crucial role in the proofs is played by suitable 'Diophantine-like' conditions (on the signature, or on the IET rotation number) which control combinatorics and exploits the classical Rauzy-Veech induction for IETs.