×

Renormalizations of circle rotations. (Russian) Zbl 1220.37032

The paper is devoted to the renormalization problem for circle maps. Given a circle rotation \(R_{\alpha}: x\mapsto x-\alpha\mod 1\) the author considers the orbit \(O_0\) of \(0\) under iterations of \(R_{\alpha}\). There exist infinitely many intervals \(I^m\) with the property that \(O_0\cap I^m\) becomes the orbit \(O_m\) for a certain rotation map \(x \mapsto x-g_m(\alpha)\mod {I^m}\). In order to study the relation between orbits \(O_0\) and \(O_m\) the author introduces the so called forward and backward renormalization functions \(R_0^m(\alpha,i)\) and \(R^{-m}_0(\alpha,i)\) related to numbers of consecutive intersections of \(O_0\) with the set \(I^m\). The author obtains closed-form expressions for these two functions and determines their asymptotics. He also derives some formulas for compositions of renormalization functions.

MSC:

37E10 Dynamical systems involving maps of the circle
37E20 Universality and renormalization of dynamical systems