Growth of groups and diffeomorphisms of the interval. (English) Zbl 1201.37060
Summary: We prove that, for all \(\alpha > 0\), every finitely generated group of \(C^{1+\alpha}\) diffeomorphisms of the interval with sub-exponential growth is almost nilpotent. Consequently, there is no group of \(C^{1+\alpha}\) interval diffeomorphisms having intermediate growth. In addition, we show that the \(C^{1+\alpha}\) regularity hypothesis for this assertion is essential by giving a \(C^1\) counter-example.
MSC:
| 37E05 | Dynamical systems involving maps of the interval |
| 20F69 | Asymptotic properties of groups |
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
