The Attractor of Renormalization and Rigidity of Towers of Critical Circle Maps
نویسنده
چکیده
We demonstrate the existence of a global attractor A with a Cantor set structure for the renormalization of critical circle mappings. The set A is invariant under a generalized renormalization transformation, whose action on A is conjugate to the twosided shift.
منابع مشابه
Hyperbolicity of Renormalization of Critical Circle Maps
The renormalization theory of critical circle maps was developed in the late 1970’s–early 1980’s to explain the occurence of certain universality phenomena. These phenomena can be observed empirically in smooth families of circle homeomorphisms with one critical point, the so-called critical circle maps, and are analogous to Feigenbaum universality in the dynamics of unimodal maps. In the works...
متن کاملRigidity and Renormalization in One Dimensional Dynamical Systems
If two smooth unimodal maps or real analytic critical circle maps have the same bounded combinatorial type then there exists a C 1+ diieomorphism conjugating the two maps along the corresponding critical orbits for some > 0. The proof is based on a detailed understanding of the orbit structure of an innnite dimensional dynamical system: the renormal-ization operator.
متن کاملAsymptotic Rigidity of Scaling Ratios for Critical Circle Mappings
Let f be a smooth homeomorphism of the circle having one cubic-exponent critical point and irrational rotation number of bounded combinatorial type. Using certain pull-back and quasi-conformal surgery techniques, we prove that the scaling ratios of f about the critical point are asymptotically independent of f . This settles in particular the golden mean universality conjecture. We introduce th...
متن کاملComplex Bounds for Renormalization of Critical Circle Maps
We use the methods developed with M. Lyubich for proving complex bounds for real quadratics to extend E. De Faria's complex a priori bounds to all critical circle maps with an irrational rotation number. The contracting property for renormalizations of critical circle maps follows. As another application of our methods we present a new proof of theorem of C. Pe-tersen on local connectivity of s...
متن کاملRenormalization Ideas in Conformal Dynamics
1. Introduction How to look at a dynamical system f at a small scale? You should take a small piece of the phase space, consider the rst return map to this piece, and then rescale it to \the original size". The new dynamical system is called the renormalization Rf of the original one. It may happen that Rf looks \similar" to f, and then you can try to repeat this procedure, and construct the se...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008