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Rotation number, invariant measures and ratio set of PL homeomorphisms of the circle. (Nombre de rotation, mesures invariantes et ratio set des homéomorphismes affines par morceaux du cercle.) (French) Zbl 1079.37033
The paper deals with piecewise affine circle homeomorphisms without periodic points. This class of maps was studied by M. Herman who found examples of transformations of this class with singular invariant measures for any rotation number. On the other hand, it is not difficult to find examples in which points of discontinuity of the derivative meet other such points and the jumps of the derivative cancel out. The present paper shows that, at least under some circumstances, this is the only way to get an absolutely continuous invariant measure, whose density then turns out to be piecewise analytic. The key condition is that that in order for the conjugacy to be piecewise \(C^1\), the number of points of non-differentiability must remain uniformly bounded for all iterates of the map. It would be interesting to know whether any of these results remain true for mappings which are piecewise smooth, but not affine, and for which the product of all jumps of the derivative is 1.

MSC:
37E10 Dynamical systems involving maps of the circle
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37E45 Rotation numbers and vectors
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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References:
[1] Dense orbits of rationals, Proc. AMS, 117, 4, 1201-1203, (1993) · Zbl 0772.54031
[2] Absolutely continuous invariant measures for a class of affine interval interval exchange maps, Proc. AMS, 123, 3533-3542, (1995) · Zbl 0848.58017
[3] Sur LES courbes définies par LES équations différentielles à la surface du tore, J. Math. Pures Appl., 11, 333-375, (1932) · JFM 58.1124.04
[4] On invariant measure for homeomorphisms of a circle with a break point, Funct. Anal. Appl., 32, 3, 153-161, (1998) · Zbl 0921.58035
[5] On groups of measure preserving transformations, Amer. J. Math., I: 81, II:85, 119-159, 551-576, (19591963) · Zbl 0087.11501
[6] Sur le problème de la génération d’une transformation donnée d’une courbe fermée, par une transformation infinitésimale, Ann. Sci. École Norm. Sup., 3e série, 67, 243-305, (1950) · Zbl 0040.15303
[7] Sur un groupe remarquable de difféomorphismes du cercle, Comm. Math. Helv., 62, 185-239, (1987) · Zbl 0647.58009
[8] Singular measures in circle dynamics, Comm. Math. Phys., 157, 213-230, (1993) · Zbl 0792.58025
[9] Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, IHÉS Publ. Math., 49, 5-233, (1979) · Zbl 0448.58019
[10] Sur LES difféomorphismes du cercle de nombre de rotation de type constant, II, 708-725, (1981) · Zbl 0501.58011
[11] Introduction to the modern theory of dynamical systems, (1995), CUP · Zbl 0878.58020
[12] Sigma-finite invariant measures for smooth mappings of the circle, J. Anal. Math., 31, 1-18, (1977) · Zbl 0346.28012
[13] The action of diffeomorphism of the circle on the Lebesgue measure, J. Anal. Math., 36, 156-166, (1979) · Zbl 0446.28016
[14] The differentiability of the conjugation of certain diffeomorphisms of the circle, Erg. Th. Dyn. Syst., 9, 643-680, (1989) · Zbl 0819.58033
[15] The absolute continuity of the conjugation of certain diffeomorphisms of the circle, Erg. Th. Dyn. Syst., 9, 681-690, (1989)
[16] On non-singular transformations of a measure space I, II, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 11, 83-97, 98-119, (1969) · Zbl 0185.11901
[17] Smoothness of conjugacies of diffeomorphisms of the circle with rotations, Russian Math. Surveys, 44, 69-99, (1989) · Zbl 0701.58053
[18] The classification of non singular actions, revisted, Erg. Th. Dyn. Syst., 11, 333-348, (1991) · Zbl 0759.58024
[19] PL homeomorphisms that are piecewise \(C^1\) conjugate to irrational rotations · Zbl 1136.37333
[20] Échanges d’intervalles affines conjugués à des linéaires, Erg. Th. Dyn. Syst., 22, 535-554, (2002) · Zbl 1043.37031
[21] Œuvres compl‘etes, 1, 137-158
[22] Il n’y a pas de contre-exemple de Denjoy analytique, C.R.A.S., série I, 298, 7, (1984) · Zbl 0573.58023
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