Changing rotation intervals of endomorphisms of the circle
Let \(E^ r_ 1(S^ 1)\) be the set of endomorphisms of the circle \(S^ 1\), of degree one and class \(C^ r\) (r\(\geq 0)\), endowed with the \(C^ r\) topology. For any \(\tilde f\in E^ r_ 1(S^ 1)\), denote by f a lift of \(\tilde f\) to \({\mathbb{R}}\). Then, following Newhouse, Palis and Takens, define the rotation set of \(\tilde f\) by \[ \rho (\tilde f)=\rho (f)=cl.\{\limsup_{n\to \infty}(f^ n(x)-x)/n| x\in {\mathbb{R}}\}. \] Denote by \(\Sigma^ r\) the subset of endomorphisms in \(E^ r_ 1(S^ 1)\) for which the rotation interval remains unchanged by \(C^ r\) small perturbations. The aim of this paper is to prove that \(\Sigma^ r\) is an open dense set in \(E^ r_ 1(S^ 1)\) for \(r\geq 1\), and to characterize the endomorphisms in \(\Sigma^ r\). The authors also characterize the endomorphisms in the boundary of a connected component of \(\Sigma^ r\).
scientific article; zbMATH DE number 3973907