×

Linearization of generalized interval exchange maps. (English) Zbl 1277.37067

Let \(I\) be an open bounded interval. A generalized interval exchange map (g.i.e.m.) \(T\) on \(I\) is defined by the following data. Let \({\mathcal A}\) be an alphabet with \(d \geq 2\) symbols. Consider “top” and “bottom” partitions of \(I\) into \(d\) open subintervals indexed by \({\mathcal A}\). More precisely, the top partition is defined by its singularity points \(u_0^t < u_1^t < \cdots < u_d^t\), where \(I=(u_0^t, u_d^t)\), and a bijection \(\pi_t : {\mathcal A} \rightarrow \{ 1,\ldots,d \}\); then \(I_{\alpha}^t = (u_{\pi_t(\alpha) -1}, u_{\pi_t(\alpha)})\) for each \(\alpha \in {\mathcal A}\) and \(I=\sqcup I^t_{\alpha}\) modulo points \(u_j^t\). The definition of the bottom partition \(I=\sqcup I^b_{\alpha}\) is similar, just replace the letter \(t\) by the letter \(b\). It is also required that \(\pi_t^{-1}(\{1,\ldots,k \}) \not= \pi_b^{-1}(\{ 1,\ldots,k \})\) for all \(k=1,\ldots,d-1\). Consider also a \((d \times d)\)-antisymmetric matrix \(\Omega\), whose element \(\Omega_{\alpha\beta}\) equals \((+1)\) if \(\pi_t(\alpha) < \pi_t(\beta)\) and \(\pi_b(\alpha) > \pi_b(\beta)\), equals \((-1)\) if \(\pi_t(\alpha) > \pi_t(\beta)\) and \(\pi_b(\alpha) < \pi_b(\beta)\), and equals \(0\) otherwise.
Then the g.i.e.m. \(T\) is defined on \(\sqcup I^t_{\alpha}\) and its restriction on each \(I^t_{\alpha}\) is an orientation-preserving homeomorphism onto the corresponding \(I^b_{\alpha}\). A g.i.e.m. \(T_0\) is “standard” if \(|I^t_{\alpha}|=|I^b_{\alpha}|\) for each \(\alpha \in {\mathcal A}\), and the restriction of \(T_0\) on each \(I^t_{\alpha}\) is a translation. We say that a g.i.e.m. \(T\) is a simple deformation of a standard i.e.m. \(T_0\) if: (i) \(T\) and \(T_0\) have the same discontinuities, (ii) \(T\) and \(T_0\) coincide in the neighborhood of each discontinuity and of the endpoints of \(I\), (iii) \(T\) is a \(C^r\)-diffeomorphism on each \(I^t_{\alpha}\). The main result of the paper is a local conjugacy theorem, which, stated in particular generality (for simple deformations), can be summarized as follows.
{Theorem.} For almost all standard i.e.m. \(T_0\) and for any integer \(r \geq 2\) amongst the \(C^{r+3}\)-simple deformations of \(T_0\), those that are \(C^r\)-conjugate to \(T_0\) by a diffeomorphism \(C^r\)-close to the identity form a \(C^1\)-submanifold of codimension \(d^*=d + (r-\tfrac{1}{2})\cdot \text{rank\,} \Omega - 2r\).

MSC:

37E05 Dynamical systems involving maps of the interval
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] V. I. Arnold, ”Small Denominators I: On the mappings of the circumference onto itself,” in Eleven Papers on Number Theory, Algebra and Functions of a Complex Variable, Providence, RI: Amer. Math. Soc., 1965, vol. 46, pp. 213-284. · Zbl 0152.41905
[2] A. Avila and A. Bufetov, ”Exponential decay of correlations for the Rauzy-Veech-Zorich induction map,” in Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, Providence, RI: Amer. Math. Soc., 2007, vol. 51, pp. 203-211. · Zbl 1149.37004
[3] A. Avila and G. Forni, ”Weak mixing for interval exchange transformations and translation flows,” Ann. of Math., vol. 165, iss. 2, pp. 637-664, 2007. · Zbl 1136.37003 · doi:10.4007/annals.2007.165.637
[4] A. Avila, S. Gouëzel, and J. Yoccoz, ”Exponential mixing for the Teichmüller flow,” Publ. Math. Inst. Hautes Études Sci., iss. 104, pp. 143-211, 2006. · Zbl 1263.37051 · doi:10.1007/s10240-006-0001-5
[5] X. Bressaud, P. Hubert, and A. Maass, ”Persistence of wandering intervals in self-similar affine interval exchange transformations,” Ergodic Theory Dynam. Systems, vol. 30, iss. 3, pp. 665-686, 2010. · Zbl 1200.37002 · doi:10.1017/S0143385709000418
[6] A. I. Bufetov, ”Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of abelian differentials,” J. Amer. Math. Soc., vol. 19, iss. 3, pp. 579-623, 2006. · Zbl 1100.37002 · doi:10.1090/S0894-0347-06-00528-5
[7] R. Camelier and C. Gutierrez, ”Affine interval exchange transformations with wandering intervals,” Ergodic Theory Dynam. Systems, vol. 17, iss. 6, pp. 1315-1338, 1997. · Zbl 0895.58019 · doi:10.1017/S0143385797097666
[8] R. de la Llave and C. Gutierrez, Absolute continuity of conjugacies among certain non-linear interval exchange transformations.
[9] G. Forni, ”Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus,” Ann. of Math., vol. 146, iss. 2, pp. 295-344, 1997. · Zbl 0893.58037 · doi:10.2307/2952464
[10] G. Forni, ”Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,” Ann. of Math., vol. 155, iss. 1, pp. 1-103, 2002. · Zbl 1034.37003 · doi:10.2307/3062150
[11] G. Forni, Sobolev regularity of solutions of the cohomological equation. · Zbl 0893.58037
[12] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Providence, R. I.: Amer. Math. Soc., 1955, vol. 36. · Zbl 0067.15204
[13] M. R. Herman, ”Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,” Inst. Hautes Études Sci. Publ. Math., vol. 49, pp. 5-233, 1979. · Zbl 0448.58019 · doi:10.1007/BF02684798
[14] M. R. Herman, ”Simple proofs of local conjugacy theorems for diffeomorphisms of the circle with almost every rotation number,” Bol. Soc. Brasil. Mat., vol. 16, iss. 1, pp. 45-83, 1985. · Zbl 0651.58008 · doi:10.1007/BF02584836
[15] M. Keane, ”Interval exchange transformations,” Math. Z., vol. 141, pp. 25-31, 1975. · Zbl 0278.28010 · doi:10.1007/BF01236981
[16] M. Keane, ”Non-ergodic interval exchange transformations,” Israel J. Math., vol. 26, iss. 2, pp. 188-196, 1977. · Zbl 0351.28012 · doi:10.1007/BF03007668
[17] M. Keane and G. Rauzy, ”Stricte ergodicité des échanges d’intervalle,” Math. Z., vol. 174, pp. 203-212, 1980. · Zbl 0479.28012 · doi:10.1007/BF01161409
[18] H. B. Keynes and D. Newton, ”A “minimal”, non-uniquely ergodic interval exchange transformation,” Math. Z., vol. 148, iss. 2, pp. 101-105, 1976. · Zbl 0308.28014 · doi:10.1007/BF01214699
[19] S. Marmi, P. Moussa, and J. -C. Yoccoz, ”The cohomological equation for Roth-type interval exchange maps,” J. Amer. Math. Soc., vol. 18, iss. 4, pp. 823-872, 2005. · Zbl 1112.37002 · doi:10.1090/S0894-0347-05-00490-X
[20] S. Marmi, P. Moussa, and J. -C. Yoccoz, ”Affine interval exchange maps with a wandering interval,” Proc. Lond. Math. Soc., vol. 100, iss. 3, pp. 639-669, 2010. · Zbl 1196.37041 · doi:10.1112/plms/pdp037
[21] H. Masur, ”Interval exchange transformations and measured foliations,” Ann. of Math., vol. 115, iss. 1, pp. 169-200, 1982. · Zbl 0497.28012 · doi:10.2307/1971341
[22] G. Rauzy, ”Échanges d’intervalles et transformations induites,” Acta Arith., vol. 34, iss. 4, pp. 315-328, 1979. · Zbl 0414.28018
[23] W. A. Veech, ”Interval exchange transformations,” J. Analyse Math., vol. 33, pp. 222-272, 1978. · Zbl 0455.28006 · doi:10.1007/BF02790174
[24] W. A. Veech, ”Gauss measures for transformations on the space of interval exchange maps,” Ann. of Math., vol. 115, iss. 1, pp. 201-242, 1982. · Zbl 0486.28014 · doi:10.2307/1971391
[25] W. A. Veech, ”The metric theory of interval exchange transformations. I. Generic spectral properties,” Amer. J. Math., vol. 106, iss. 6, pp. 1331-1359, 1984. · Zbl 0631.28006 · doi:10.2307/2374396
[26] W. A. Veech, ”The metric theory of interval exchange transformations. II. Approximation by primitive interval exchanges,” Amer. J. Math., vol. 106, iss. 6, pp. 1361-1387, 1984. · Zbl 0631.28007 · doi:10.2307/2374397
[27] J. Yoccoz, ”Continued fraction algorithms for interval exchange maps: an introduction,” in Frontiers in Number Theory, Physics, and Geometry. I, New York: Springer-Verlag, 2006, pp. 401-435. · Zbl 1127.28011 · doi:10.1007/978-3-540-31347-2_12
[28] J. Yoccoz, Cours 2005 : Échange d’intervalles.
[29] J. -C. Yoccoz, ”Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne,” Ann. Sci. École Norm. Sup., vol. 17, iss. 3, pp. 333-359, 1984. · Zbl 0595.57027
[30] J. Yoccoz, ”Interval exchange maps and translation surfaces,” in Homogeneous Flows, Moduli Spaces and Arithmetic, Providence, RI: Amer. Math. Soc., 2010, vol. 10, pp. 1-69. · Zbl 1248.37038
[31] A. Zorich, ”Flat surfaces,” in Frontiers in Number Theory, Physics, and Geometry. I, New York: Springer-Verlag, 2006, pp. 437-583. · Zbl 1129.32012 · doi:10.1007/3-540-31347-8_13
[32] A. Zorich, ”Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 46, iss. 2, pp. 325-370, 1996. · Zbl 0853.28007 · doi:10.5802/aif.1517
[33] A. Zorich, ”Deviation for interval exchange transformations,” Ergodic Theory Dynam. Systems, vol. 17, iss. 6, pp. 1477-1499, 1997. · Zbl 0958.37002 · doi:10.1017/S0143385797086215
[34] A. Zorich, ”How do the leaves of a closed \(1\)-form wind around a surface?,” in Pseudoperiodic Topology, Providence, RI: Amer. Math. Soc., 1999, vol. 197, pp. 135-178. · Zbl 0976.37012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.