2012/05/09 by John Milnor, Milnor, John, Alfredo Poirier +1 · 2 citations
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematics and Applications #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1205.2668
openalex publication_date 2012/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider polynomial maps f:\C→\C of degree d≥ 2, or more generally polynomial maps from a finite union of copies of \C to itself. In the space of suitably normalized maps of this type, the hyperbolic maps form an open set called the hyperbolic locus. The various connected components of this hyperbolic locus are called hyperbolic components, and those hyperbolic components with compact closure (or equivalently those contained in the "connectedness locus") are called bounded hyperbolic components. It is shown that each bounded hyperbolic component is a topological cell containing a unique post-critically finite map called its center point. For each degree d, the bounded hyperbolic components can be separated into finitely many distinct types, each of which is characterized by a suitable reduced mapping scheme Sf. Any two components with the same reduced mapping scheme are canonically biholomorphic to each other. There are similar statementsfor real polynomial maps, for polynomial maps with marked critical points, and for rational maps. Appendix A, by Alfredo Poirier, proves that every reduced mapping scheme can be represented by some classical hyperbolic component, made up of polynomial maps of \C. This paper is a revised version of [M2], which was circulated but not published in 1992.