Changing rotation intervals of endomorphisms of the circle

Rotation intervals for a family of degree one circle maps

“…On the assumption that f k is never a homeomorphism it is shown that if p+(/ A )eQ then there is an e > 0 such that either P+(U)^P+(fx) for all fi€(X-e,\ + e) or else p+(/ M )=sp+(/ A ) for all p.e (A-e, A + e). This theorem is a slight generalization of results obtained by Bamon, Malta and Pacifico in [2], and it serves to set the stage for the other two main theorems.…”

“…On the assumption that f k is never a homeomorphism it is shown that if p+(/ A )eQ then there is an e > 0 such that either P+(U)^P+(fx) for all fi€(X-e,\ + e) or else p+(/ M )=sp+(/ A ) for all p.e (A-e, A + e). This theorem is a slight generalization of results obtained by Bamon, Malta and Pacifico in [2], and it serves to set the stage for the other two main theorems.Theorems 3.5, 3.7, and 3.8 deal with a more specific family R x °f. In addition they require that we impose several differentiability conditions on / Specifically we assume that / is C 3 with precisely two critical points, a quadratic local minimum and a quadratic local maximum, and that / has a negative Schwarzian derivative.…”

“…Although the differences between the conditions presented in Proposition 2.5 and those presented in [2] are very slight, the latter are not in fact sufficient for C°-persistence, as the following example will show. …”

“…The conditions obtained in [2] for C 1 -persistence of p+(F) are necessary but not quite sufficient for C°-persistence. We have instead the following parallel result.…”

Realizing rotation numbers on annular continua

“…The success of this rotational invariant in the study of circle homeomorphisms led to a vast literature of generalizations and applications of this notion in different settings, for instance for endomorphisms of the circle [Ito81, NPT83,BMP86], homeomorphisms of the torus [MZ89, LM91,Fra89], and surfaces of higher genus [Pol92,Fra95].…”