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Dynamics for rotation pseudogroups. (La dynamique des pseudogroupes de rotations.) (French) Zbl 0791.58055
We study dynamical systems on the circle generated by a finite number of partially defined rotations. We construct new examples with all orbits dense (this leads to non-simplicial free actions of free groups on \(\mathbb R\)-trees). We study the generic dynamics for these pseudogroups and their 1-parameter families. We show that, in suitable 2-parameter families, the set of pseudogroups having a dense orbit is a Sierpiński curve. We generalize results on interval exchange transformations obtained by Boshernitzan, Veech, Rips.

37E10 Dynamical systems involving maps of the circle
37E45 Rotation numbers and vectors
22A22 Topological groupoids (including differentiable and Lie groupoids)
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
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[1] [AL] Arnoux, P., Levitt, G.: Sur l’unique ergodicité des 1-formes fermées singulières. Invent. Math.84, 141-156 (1986) · Zbl 0577.58021
[2] [Bo] Boshernitzan, M.: Rank two interval exchange transformations. Ergodic Theory Dyn. Syst.8, 379-394 (1988) · Zbl 0657.28013
[3] [E] Ehresmann, C.: Sur la théorie des espaces fibrés. In: Coll. Int. Top. Alg., pp. 3-15. Paris: CNRS 1947. In: ?uvres complètes, I, pp. 133-145. Amiens 1984
[4] [Ha1] Haefliger, A.: Some remarks on foliations with minimal leaves. J. Differ. Geom.15, 269-284 (1980) · Zbl 0444.57016
[5] [Ha2] Haefliger, A.: Groupoïdes d’holonomie et classifiants. In: Structures transverses des feuilletages. (Astérisque, vol. 116, pp. 70-97) Paris: Soc. Math. Fr. 1984 · Zbl 0562.57012
[6] [Ha3] Haefliger, A.: Pseudogroups of local isometries. In: Cordero (ed.) Differential Geometry, Santiago de Compostela. (Pitman Res. Notes Math. Ser., vol. 131, pp. 174-197) Harlow: Longman 1985
[7] [Ha4] Haefliger, A.: Leaf closures in Riemannian foliations. In: Matsumoto, Y. et al. (eds.) A fete of topology, papers dedicated to I. Tamura, pp. 3-32. Boston, MA: Academic Press 1988
[8] [Im] Imanishi, H.: On codimension one foliations defined by closed one forms with singularities. J. Math. Kyoto Univ.19, 285-291 (1979) · Zbl 0417.57010
[9] [Ke] Keane, M.: Interval exchange transformations. Math. Z.141, 25-31 (1975) · Zbl 0288.28020
[10] [Le1] Levitt, G.: Pantalons et feuilletages des surfaces. Topology21, 9-33 (1982) · Zbl 0473.57014
[11] [Le2] Levitt, G.: Sur les mesures transverses invariantes d’un feuilletage de codimension 1. C.R. Acad. Sci., Paris290, 1139-1140 (1980) · Zbl 0459.57017
[12] [Le3] Levitt, G.: 1-formes fermées singulières et groupe fondamental. Invent. Math.88, 635-667 (1987) · Zbl 0594.57014
[13] [Le4] Levitt, G.: Groupe fondamental de l’espace des feuilles dans les feuilletages sans holonomie. J. Differ. Geom.31, 711-761 (1990) · Zbl 0714.57016
[14] [Le5] Levitt, G.: Constructing free actions on 670-1. Duke Math. J.69, 615-633 (1993) · Zbl 0794.57001
[15] [Mo] Molino, P.: Riemannian foliations. (Prog. Math., vol. 73) Boston Basel Stuttgart: Birkhäuser 1988 · Zbl 0824.53028
[16] [Ra] Rauzy, G.: Une généralisation du développement en fraction continue. Sémin. Delange-Pisot-Poitou, n0 15 fasc. 1, Paris, 18ème année (1976-1977)
[17] [Sac] Sacksteder, R.: Foliations and pseudogroups. Am. J. Math.87, 79-102 (1965) · Zbl 0136.20903
[18] [Sa1] Salem, E.: Une généralisation du théorème de Myers-Steenrod aux pseudogroupes d’isométries. Ann. Inst. Fourier38, 185-200 (1988)
[19] [Sal2] Salem, E.: Riemannian foliations and pseudogroups. In: Molino, P.: Riemannian foliations, appendix D. (Prog. Math., vol. 73) Boston Basel Stuttgart: Birkhäuser 1988
[20] [Si1] Sierpi?ski, W.: Sur une courbe dont tout point est un point de ramification. C.R. Acad. Sci., Paris160, 302-305 (1915); ?uvres choisies II, pp. 99-106. Varsovie: PWN 1975 · JFM 45.0628.02
[21] [Si2] Sierpi?ski, W.: Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée. C.R. Acad. Sci., Paris162, 629-632 (1916); ?uvres choisies II, pp. 107-119. Varsovie: PWN 1975 · JFM 46.0295.02
[22] [SS] Singer, I.M., Sternberg, S.: The infinite groups of Lie and Cartan. J. Anal. Math.15. 1-114 (1965) · Zbl 0277.58008
[23] [VW] Veblen, O., Whitehead, J.H.C.: The foundations of differential geometry. Camb. Tracts Math.29 (1932) · Zbl 0005.21801
[24] [Ve1] Veech, W.A.: Interval exchange transformations. J. Anal. Math.33, 222-272 (1978) · Zbl 0455.28006
[25] [Ve2] Veech, W.A.: Boshernitzan’s criterion for unique ergodicity of an interval exchange transformation. Ergodic Theory Dyn. Sys.7, 149-153 (1987) · Zbl 0657.28012
[26] [Wh] Whyburn, G.T.: Topological characterization of the Sierpi?ski curve. Fundam. Math.45, 320-324 (1958) · Zbl 0081.16904
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