Periodic attractors of perturbed one-dimensional~maps (Q2873995)

The author of this paper considers the question of how many periodic attractors there can be for maps in a neighborhood of a given map. The basic setting is the following: \({\mathcal N}\) is an interval or a circle, and \(f:{\mathcal N}\to{\mathcal N}\) is a \(C^\infty\) map. A closed interval \(I\subset{\mathcal N}\) is called periodic if there is an integer \(n\) such that \(f^n(I)= I\) and \(f^n:I\to I\) is a bijection. A ``pack of periodic points'' is a collection of periodic points belonging to a closed periodic interval for which there is no larger periodic interval that contains more periodic points. A ``pack of periodic orbits'' is defined in an obvious way.NEWLINENEWLINE The main result is the following: If \(f:{\mathcal N}\to{\mathcal N}\) is a \(C^\infty\) map with non-flat critical points, then there exist a \(C^\infty\) neighborhood \({\mathcal F}\) of \(f\) and positive \(M\) and \(p\) such that for any \(g\in{\mathcal F}\), there are at most \(M\) exceptional packs of periodic orbits, and if \(p\) is a periodic point of \(g\) which is not a member of any of these exceptional packs, then \(|Dg^n(g)|> 1+p\) where \(n\) is a period of \(p\).NEWLINENEWLINE The proof relies on a theorem of Singer on periodic maps with negative Schwarzian derivative, and a uniform contraction principle. The possibility of degenerate critical points makes for a significant complication.