Natalia Goncharuk: "Complex rotation numbers and renormalization ": *annulé*

mercredi 15 déc. 2021, 14:00 → 15:00 Europe/Paris

435 (UMPA)

Given an analytic circle diffeomorphism $f$ and a complex number $\omega\in \bbH$, consider a fundamental domain of $f+\omega$ in $\bbC/\bbZ$ that is close to $\bbR/\bbZ$. The quotient space of this fundamental domain via $f+\omega$ is a torus.  The complex rotation number​ of $f+\omega$ equals the modulus of this torus. This construction is due to V. Arnold (1978).

As $\omega\to 0$, the limit values of the complex rotation number form an approximately self-similar  set ``bubbles''.

 

I will give a survey of results on complex rotation numbers and bubbles, discuss a new hyperbolicity result for the renormalization of analytic diffeomorphisms $f\colon \bbR/\bbZ \to \bbC/\bbZ$ (joint work with M. Yampolsky) and the implications of this new result to the geometry of bubbles (joint work with I. Gorbovickis).