Bifurcation to badly ordered orbits in one-parameter families of circle maps, or angels fallen from the devil’s staircase

Abstract

We discuss the structure of the bifurcation set of a one-parameter family of endomorphisms of S 1 {S^1} having two critical points and negative Schwarzian derivative. We concentrate on the case in which one of the endpoints of the rotation set is rational, providing a partial characterization of components of the nonwandering set having specified rotation number and the bifurcations in which they are created. In particular we find, for each rational rotation number p ′ / q ′ p’/q’ less than the upper boundary of the rotation set p / q p/q , infinitely many saddle-node bifurcations to badly ordered periodic orbits of rotation number p ′ / q ′ p’/q’ .