Computation of rotation numbers for a class of PL-circle homeomorphisms

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October 2012, 32(10): 3399-3419. doi: 10.3934/dcds.2012.32.3399

Computation of rotation numbers for a class of PL-circle homeomorphisms

Abdelhamid Adouani ^1, and Habib Marzougui ^1,

1.	University of Carthage, Faculty of Science of Bizerte, Department of Mathematics, Zarzouna, 7021, Tunisia, Tunisia

Received May 2011 Revised February 2012 Published May 2012

Abstract
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We give an explicite formula to compute rotation numbers of piecewise linear (PL) circle homeomorphisms $f$ with the product of $f$-jumps in the break points contained in a same orbit is trivial. In particular, a simple formulas are then given for particular PL-homeomorphisms such as the PL-Herman's examples. We also deduce that if the slopes of $f$ are integral powers of an integer $m\geq 2$ and break points and their images under $f$ are $m$-adic rational numbers, then the rotation number of $f$ is rational.

Keywords: conjugacy, jump., break point, class P circle homeomorphism, rotation number, Piecewise linear circle homeomorphism.

Mathematics Subject Classification: Primary: 37C15, 37E10, 37E4.

Citation: Abdelhamid Adouani, Habib Marzougui. Computation of rotation numbers for a class of PL-circle homeomorphisms. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3399-3419. doi: 10.3934/dcds.2012.32.3399

References:

[1]	A. Adouani and H. Marzougui, Sur les Homéomorphismes du cercle de classe $P$ $C^r$ par morceaux ($r\geq 1$) qui sont conjugués $C^r$ par morceaux aux rotations irrationnelles, Ann. Inst. Fourier (Grenoble), 58 (2008), 755-775. doi: 10.5802/aif.2368.
[2]	A. Adouani and H. Marzougui, On piecewise smoothness of conjugacy of class P circle homeomorphisms to diffeomorphisms and rotations, Dynamical Systems, to appear, 2012.
[3]	M. D. Boshernitzan, Dense orbits of rationals, Proc. Amer. Math. Soc., 117 (1993), 1201-1203. doi: 10.1090/S0002-9939-1993-1134622-6.
[4]	A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl., 11 (1932), 333-375.
[5]	Z. Coelho, A. Lopez and L. F. da Rocha, Absolutely continuous invariant measures for a class of affine interval exchange maps, Proc. Amer. Math. Soc., 123 (1995), 3533-3542. doi: 10.1090/S0002-9939-1995-1322918-6.
[6]	M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5-233. doi: 10.1007/BF02684798.
[7]	I. Liousse, PL Homeomorphisms of the circle which are piecewise $C^1$ conjugate to irrational rotations, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 269-280. doi: 10.1007/s00574-004-0014-y.
[8]	I. Liousse, Rotation numbers in Thompson-Stein groups and applications, Geom. Dedicata, 131 (2008), 49-71. doi: 10.1007/s10711-007-9216-y.
[9]	I. Liousse, Nombre de rotation dans les groupes de Thompson généralisés, automorphismes, preprint, 2006. Available from: http://hal.ccsd.cnrs.fr/ccsd-00004554.
[10]	H. Poincaré, Oeuvres complètes, t.1 (1885), 137-158.

show all references

References:

[1]	A. Adouani and H. Marzougui, Sur les Homéomorphismes du cercle de classe $P$ $C^r$ par morceaux ($r\geq 1$) qui sont conjugués $C^r$ par morceaux aux rotations irrationnelles, Ann. Inst. Fourier (Grenoble), 58 (2008), 755-775. doi: 10.5802/aif.2368.
[2]	A. Adouani and H. Marzougui, On piecewise smoothness of conjugacy of class P circle homeomorphisms to diffeomorphisms and rotations, Dynamical Systems, to appear, 2012.
[3]	M. D. Boshernitzan, Dense orbits of rationals, Proc. Amer. Math. Soc., 117 (1993), 1201-1203. doi: 10.1090/S0002-9939-1993-1134622-6.
[4]	A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl., 11 (1932), 333-375.
[5]	Z. Coelho, A. Lopez and L. F. da Rocha, Absolutely continuous invariant measures for a class of affine interval exchange maps, Proc. Amer. Math. Soc., 123 (1995), 3533-3542. doi: 10.1090/S0002-9939-1995-1322918-6.
[6]	M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5-233. doi: 10.1007/BF02684798.
[7]	I. Liousse, PL Homeomorphisms of the circle which are piecewise $C^1$ conjugate to irrational rotations, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 269-280. doi: 10.1007/s00574-004-0014-y.
[8]	I. Liousse, Rotation numbers in Thompson-Stein groups and applications, Geom. Dedicata, 131 (2008), 49-71. doi: 10.1007/s10711-007-9216-y.
[9]	I. Liousse, Nombre de rotation dans les groupes de Thompson généralisés, automorphismes, preprint, 2006. Available from: http://hal.ccsd.cnrs.fr/ccsd-00004554.
[10]	H. Poincaré, Oeuvres complètes, t.1 (1885), 137-158.

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