Further scaling properties for circle maps
Universal scaling properties for circle maps which for rotation numbers \(\rho\) algebraic of degree two have been found recently by several groups are generalized to actions of the group \({\mathbb{Z}}^{k-1}\) on the circle. If the rotation numbers of their generators \(f_ 1,...,f_{k-1}\) are fixed or periodic points of the Perron-Jacobi algorithm and are therefore algebraic numbers of degree \(k>2\) then the corresponding action shows a universal behaviour under certain iterations independent of the special form of the \(f_ i's\). A renormalization group transformation is established which accounts for this behaviour and which in the case \(k=2\) reduces exactly to the known transformation for circle maps. A more elaborate version of this work has appeared in J. Stat. Phys. 38, 785-803 (1985).
\({\mathbb{Z}}^ h\) actions on the circle
Universal scaling properties for circle maps
scientific article; zbMATH DE number 3927955