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On one-dimensional dynamics in intermediary regularity. (Sur la dynamique unidimensionnelle en régularité intermédiaire.) (French) Zbl 1139.37025

The authors obtain several new results on the dynamics of diffeomorphism groups of one-dimensional Hausdorff manifolds which are “of intermediate regularity”, that is, whose differentiability is at least 1 but strictly less than 2. Motivation of such a study can be found in work of Milnor, Sergeraert and Herman, continuing work of Denjoy on rotation numbers of diffeomorphisms of the circle, and in the work of Hurder, Katok and others on Godbillon-Vey invariants of foliations.
The authors use methods of probabilistic nature. They first obtain the following three general results on abelian group actions on the interval and the circle which are of intermediate regularity.
Theorem A. If \(d\) is an integer \(\geq 2\) and \(\epsilon >0\), then any free \(\mathbb{Z}\)-action on the circle by diffeomorphisms of class \(C^{1+1/d+\epsilon}\) is minimal.
Theorem A had been conjectured by Tsuboi, and the case \(d=1\) is a sort of generalization of a result of Denjoy.
Theorem B. Let \(d\) be an integer \(\geq 2\) and \(\epsilon >0\). Let \(f_1,\ldots,f_{d+1}\) be commuting diffeomorphisms of class \(C^1\) of the interval \([0,1]\). Suppose there exist open disjoint intervals \(I_{n_{1},\dots,n_{d}}\) contained in \((0,1)\) such that this inclusion respects the lexicographic ordering, and such that, for all \((n_1,\dots,n_d)\in\mathbb{Z}^d\) and for all \(i\in\{1,\dots,d\}\), \[ f_i(I_{n_{1}\ldots,n_{i},\ldots,n_{d}})=I_{n_{1}\ldots,n_{i-1},\dots,n_{d}}, \]
and such that
\[ f_{d+1}(I_{n_{1}\dots,n_{d}})=I_{n_{1}\ldots,n_{d}}. \]
If furthermore \(f_1,\dots,l_d\) are of class \(C^{1+1/d+\epsilon}\), then the restriction of \(f_{d+1}\) to the union of intervals \(I_{n_{1}\dots,n_{i},\dots,n_{d}}\) is the identity.
Theorem C. Let \(\Gamma\) be a subgroup of \(\mathrm{Diff}_+^{1+\tau}([0,1])\) isomorphic to \(\mathbb{Z}^d\), with \(\tau>1/d\) and \(d\geq 2\). If the restriction of the action of \(\Gamma\) to \((0,1)\) is free, then it is minimal and topologically conjugate to the action of a translation group.
The authors also obtain other results such as the following:
Theorem D. Any countable subgroup of orientation-preserving homeomorphims of the circle or of the interval \([0,1]\) is topologically conjugate to a group of Lipschitz homeomorphisms.
The next result uses the notion of resilient orbit for a pseudo-group of local homeomorphisms of \(X\). An orbit \(O(x)\) of a point \(x\in X\) is said to be resilient if there exists \(g\in\Gamma\) whose domain contains an interval \([x,p)\) or \((p,x]\) for which \(x\) is an attractive topological fixed point, and, furthermore, \(O(x)\) intersects the interior of this interval.
Theorem E. Let \(\Gamma\) be a pseudo-group of orientation-preserving homeomorphisms of a compact Hausdorff one-dimensional manifold \(X\). If \(\Gamma\) does not preserve any probability measure on \(X\), then \(\Gamma\) has resilient orbits. If furthermore the elements of \(\Gamma\) are local diffeomorphisms of class \(C^1\), then \(\Gamma\) has orbits resilient hyperbolic orbits.
The following theorem generalizes a theorem of S. Hurder:
Theorem F. If \(\Gamma\) is a subgroup of \(\mathrm{Diff}^1_+(S^1)\) that preserves no probability measure on the circle, then \(\Gamma\) contains elements that have only hyperbolic fixed points.

MSC:

37E05 Dynamical systems involving maps of the interval
22F05 General theory of group and pseudogroup actions
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