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Persistence of wandering intervals in self-similar affine interval exchange transformations

Published online by Cambridge University Press:  23 June 2009

XAVIER BRESSAUD
Affiliation:
Université Paul Sabatier, Institut de Mathématiques de Toulouse, F-31062 Toulouse Cedex, France (email: bressaud@math.univ-toulouse.fr)
PASCAL HUBERT
Affiliation:
Laboratoire Analyse, Topologie et Probabilités, Case cour A, Faculté des Sciences de Saint-Jerôme, Avenue Escadrille Normandie-Niemen, 13397 Marseille Cedex 20, France (email: hubert@cmi.univ-mrs.fr)
ALEJANDRO MAASS
Affiliation:
Centro de Modelamiento Matemático and Departamento de Ingeniería Matemática, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile (email: amaass@dim.uchile.cl)

Abstract

In this article we prove that given a self-similar interval exchange transformation T(λ,π), whose associated matrix verifies a quite general algebraic condition, there exists an affine interval exchange transformation with wandering intervals that is semi-conjugated to it. That is, in this context the existence of Denjoy counterexamples occurs very often, generalizing the result of Cobo [Piece-wise affine maps conjugate to interval exchanges. Ergod. Th. & Dynam. Sys.22 (2002), 375–407].

Type
Research Article
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Adamczewski, B.. Symbolic discrepancy and self-similar dynamics. Ann. Inst. Fourier (Grenoble) 54 (2004), 22012234.CrossRefGoogle Scholar
[2]Avila, A. and Viana, M.. Simplicity of Lyapunov spectra: proof of the Zorich–Kontsevich conjecture. Acta Math. 198 (2007), 156.CrossRefGoogle Scholar
[3]Canterini, V. and Siegel, A.. Automate des préfixes-suffixes associé une substitution primitive [Prefix–suffix automaton associated with a primitive substitution]. J. Théor. Nombres Bordeaux 13(2) (2001), 353369.CrossRefGoogle Scholar
[4]Camelier, R. and Gutierrez, C.. Affine interval exchange transformations with wandering intervals. Ergod. Th. & Dynam. Sys. 17 (1997), 13151338.CrossRefGoogle Scholar
[5]Cobo, M.. Piece-wise affine maps conjugate to interval exchanges. Ergod. Th. & Dynam. Sys. 22 (2002), 375407.CrossRefGoogle Scholar
[6]Denjoy, A.. Sur les courbes definies par les équations differentielles à la surface du tore. J. Math. Pures Appl. 11(9) (1932), 333375.Google Scholar
[7]Dumont, J. M. and Thomas, A.. Digital sum moments and substitutions. Acta Arith. 64 (1993), 205225.CrossRefGoogle Scholar
[8]Dumont, J. M. and Thomas, A.. Digital sum problems and substitutions on a finite alphabet. J. Number Theory 39(3) (1991), 351366.CrossRefGoogle Scholar
[9]Fogg, P.. Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics, 1794). Springer, Berlin, 2002.CrossRefGoogle Scholar
[10]Gjerde, R. and Johansen, O.. Bratteli–Vershik models for Cantor minimal systems associated to interval exchange transformations. Math. Scand. 90 (2002), 87100.CrossRefGoogle Scholar
[11]Halász, G.. Remarks on the remainder in Birkhoff’s ergodic theorem. Acta Math. Acad. Sci. Hungar. 28 (1976), 389395.CrossRefGoogle Scholar
[12]Herman, R. H., Putnam, I. and Skau, C. F.. Ordered Bratteli diagrams, dimension groups and topological dynamics. Internat. J. Math. 3 (1992), 827864.CrossRefGoogle Scholar
[13]Hubert, P. and Lanneau, E.. Veech groups without parabolic elements. Duke Math. J. 133(2) (2006), 335346.CrossRefGoogle Scholar
[14]Kenyon, R. and Smillie, J.. Billiards on rational-angled triangles. Comment. Math. Helv. 75 (2000), 65108.CrossRefGoogle Scholar
[15]Levitt, G.. La décomposition dynamique et la différentiabilité des feuilletages des surfaces. Ann. Inst. Fourier (Grenoble) 37 (1987), 85116.CrossRefGoogle Scholar
[16]Marmi, S., Moussa, P. and Yoccoz, J. C.. The cohomological equation for Roth-type interval exchange maps. J. Amer. Math. Soc. 18(4) (2005), 823872.CrossRefGoogle Scholar
[17]Marmi, S., Moussa, P. and Yoccoz, J. C.. Affine interval maps with a wandering interval. Preprint, 2008, arXiv:0805.4737v2.Google Scholar
[18]Queffélec, M.. Substitution Dynamical Systems—Spectral Analysis (Lecture Notes in Mathematics, 1294). Springer, Berlin, 1987.CrossRefGoogle Scholar
[19]Rauzy, G.. Echanges d’intervalles et transformations induites. Acta Arith. 34 (1979), 315328.CrossRefGoogle Scholar
[20]Thurston, W.. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. 19 (1988), 417431.CrossRefGoogle Scholar
[21]Veech, W. A.. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115 (1982), 201242.CrossRefGoogle Scholar
[22]Yoccoz, J. C.. Continuous fraction algorithms for interval exchange maps: an introduction. Frontiers in Number Theory, Physics and Geometry, Vol. I. On Random matrices, Zeta Functions and Dynamical Systems. Eds. Cartier, P., Julia, B., Moussa, P. and Vanhove, P.. Springer, Berlin, 2006, pp. 403437.Google Scholar
[23]Zorich, A.. Finite Gauss measure on the space of interval exchange transformations, Lyapunov exponents. Ann. Inst. Fourier (Grenoble) 46(2) (1996), 325370.CrossRefGoogle Scholar
[24]Zorich, A.. Flat surfaces. Frontiers in Number Theory, Physics and Geometry, Vol. I. On Random Matrices, Zeta Functions and Dynamical Systems. Eds. Cartier, P., Julia, B., Moussa, P. and Vanhove, P.. Springer, Berlin, 2006, pp. 439585.Google Scholar