## Rigidity of critical circle maps.(English)Zbl 1400.37047

A critical circle map is an orientation-preserving $$C^4$$ circle homeomorphism $$f$$ with exactly one critical point $$c$$ such that, in a neighborhood of $$c$$, the map $$f$$ can be written as $$f(t)=f(c)+(\phi (t))^{2d+1}$$ where $$\phi$$ is a $$C^4$$ local diffeomorphism with $$\phi (c)=0$$ and $$d \in \mathbb{N}$$ with $$d\geq 1$$. The criticality (also called order, type or exponent ) of the critical point $$c$$ is the odd integer $$2d+1$$. The critical point is also called nonflat.
A $$C^4$$ critical circle map $$f$$ with irrational rotation number generates a sequence $$\{\mathcal{R}^n(f)\}_{n \in \mathbb{N}}$$ of commuting pairs of interval maps, each such map being the renormalization of the previous one. The authors observe that a critical commuting pair is a special case of a generalized interval exchange map of two intervals and the renormalization operator is the restriction of the Zorich accelerated version of the Rauzy-Veech renormalization for interval exchange maps.
The main result in the paper states that there exists a universal constant $$\lambda$$, a point in the open interval $$(0,\,1),$$ such that given two $$C^4$$ critical circle maps $$f$$ and $$g$$ with the same irrational rotation number and the same criticality, there exists $$C >0$$ (a constant that depends on $$f$$ and $$g$$) such that for all $$n \in \mathbb{N}$$ we have $$d_2 (\mathcal{R}^n(f), \mathcal{R}^n(g)) \leq C \lambda^n,$$ where $$d_2$$ denotes the $$C^2$$-distance in the space of $$C^2$$ critical commuting pairs.
This theorem implies the following rigidity theorem: given two $$C^4$$ circle homeomorphisms $$f$$ and $$g$$ with the same irrational rotation number and with a unique critical point of the same odd type. If $$h$$ denotes the unique topological conjugacy between $$f$$ and $$g$$ that maps the critical point of $$f$$ to the critical point of $$g$$, then the following three statements hold: (1) $$h$$ is a $$C^1$$-diffeomorphism; (2) $$h$$ is a $$C^{1+\alpha}$$ at the critical point of $$f$$ for a universal $$\alpha >0$$; and (3) for a full Lebesgue measure set of rotation numbers, $$h$$ is a $$C^{1+\alpha}$$-diffeomorphism.
This paper has 13 sections, an appendix and a list of references.

### MSC:

 3.7e+11 Dynamical systems involving maps of the circle 3.7e+06 Dynamical systems involving maps of the interval 3.7e+21 Universality and renormalization of dynamical systems 3.7e+46 Rotation numbers and vectors

### Keywords:

critical circle map; smooth rigidity; renormalization
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### References:

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