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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

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 & 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.  Google Scholar

[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. Google Scholar

[3]

M. D. Boshernitzan, Dense orbits of rationals, Proc. Amer. Math. Soc., 117 (1993), 1201-1203. doi: 10.1090/S0002-9939-1993-1134622-6.  Google Scholar

[4]

A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl., 11 (1932), 333-375. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

I. Liousse, Rotation numbers in Thompson-Stein groups and applications, Geom. Dedicata, 131 (2008), 49-71. doi: 10.1007/s10711-007-9216-y.  Google Scholar

[9]

I. Liousse, Nombre de rotation dans les groupes de Thompson généralisés, automorphismes, preprint, 2006. Available from: http://hal.ccsd.cnrs.fr/ccsd-00004554. Google Scholar

[10]

H. Poincaré, Oeuvres complètes, t.1 (1885), 137-158. Google Scholar

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.  Google Scholar

[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. Google Scholar

[3]

M. D. Boshernitzan, Dense orbits of rationals, Proc. Amer. Math. Soc., 117 (1993), 1201-1203. doi: 10.1090/S0002-9939-1993-1134622-6.  Google Scholar

[4]

A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl., 11 (1932), 333-375. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

I. Liousse, Rotation numbers in Thompson-Stein groups and applications, Geom. Dedicata, 131 (2008), 49-71. doi: 10.1007/s10711-007-9216-y.  Google Scholar

[9]

I. Liousse, Nombre de rotation dans les groupes de Thompson généralisés, automorphismes, preprint, 2006. Available from: http://hal.ccsd.cnrs.fr/ccsd-00004554. Google Scholar

[10]

H. Poincaré, Oeuvres complètes, t.1 (1885), 137-158. Google Scholar

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