Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics

Authors
• 1 Kyoto University, Department of Mathematics, Kitashirakawa-Oiwakecho, Kyoto, 606-8502, Japan , Kyoto (Japan)
• 2 Hitotsubashi University, Graduate School of Commerce and Management, 2-1 Naka, Kunitachi, Tokyo, 186-8601, Japan , Tokyo (Japan)
• 3 Imperial College London, Department of Mathematics, 180 Queen’s Gate, London, UK , London (United Kingdom)
• 4 Lobachevsky University of Nizhny Novgorod, Nizhny Novgorod, 603950, Russia , Nizhny Novgorod (Russia)
Type
Published Article
Journal
Mathematische Annalen
Publisher
Springer Berlin Heidelberg
Publication Date
Sep 03, 2016
Volume
368
Issue
3-4
Pages
1277–1309
Identifiers
DOI: 10.1007/s00208-016-1468-0
Source
Springer Nature
Keywords
We consider semigroup actions on the unit interval generated by strictly increasing Cr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^r$$\end{document}-maps. We assume that one of the generators has a pair of fixed points, one attracting and one repelling, and a heteroclinic orbit that connects the repeller and attractor. We also assume that the other generators form a robust blender, which can bring the points from a small neighborhood of the attractor to an arbitrarily small neighborhood of the repeller. This is a model setting for partially hyperbolic systems with one central direction. We show that, under additional conditions on f′′f′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{f''}{f'}$$\end{document} and the Schwarzian derivative, the above semigroups exhibit, Cr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^r$$\end{document}-generically for any r≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \ge 3$$\end{document}, arbitrarily fast growth of the number of periodic points as a function of the period. We also show that a Cr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^r$$\end{document}-generic semigroup from the class under consideration supports an ultimately complicated behavior called universal dynamics.