# Eigenvalues and simplicity of interval exchange transformations

Sébastien Ferenczi; Luca Q. Zamboni

Annales scientifiques de l'École Normale Supérieure (2011)

- Volume: 44, Issue: 3, page 361-392
- ISSN: 0012-9593

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topFerenczi, Sébastien, and Zamboni, Luca Q.. "Eigenvalues and simplicity of interval exchange transformations." Annales scientifiques de l'École Normale Supérieure 44.3 (2011): 361-392. <http://eudml.org/doc/272107>.

@article{Ferenczi2011,

abstract = {For a class of $d$-interval exchange transformations, which we call the symmetric class, we define a new self-dual induction process in which the system is successively induced on a union of sub-intervals. This algorithm gives rise to an underlying graph structure which reflects the dynamical behavior of the system, through the Rokhlin towers of the induced maps. We apply it to build a wide assortment of explicit examples on four intervals having different dynamical properties: these include the first nontrivial examples with eigenvalues (rational or irrational), the first ever example of an exchange on more than three intervals satisfying Veech’s simplicity (though this weakening of the notion of minimal self-joinings was designed in 1982 to be satisfied by interval exchange transformations), and an unexpected example which is non uniquely ergodic, weakly mixing for one invariant ergodic measure but has rational eigenvalues for the other invariant ergodic measure.},

author = {Ferenczi, Sébastien, Zamboni, Luca Q.},

journal = {Annales scientifiques de l'École Normale Supérieure},

keywords = {interval exchanges; self-dual induction; eigenvalues; Veech simplicity},

language = {eng},

number = {3},

pages = {361-392},

publisher = {Société mathématique de France},

title = {Eigenvalues and simplicity of interval exchange transformations},

url = {http://eudml.org/doc/272107},

volume = {44},

year = {2011},

}

TY - JOUR

AU - Ferenczi, Sébastien

AU - Zamboni, Luca Q.

TI - Eigenvalues and simplicity of interval exchange transformations

JO - Annales scientifiques de l'École Normale Supérieure

PY - 2011

PB - Société mathématique de France

VL - 44

IS - 3

SP - 361

EP - 392

AB - For a class of $d$-interval exchange transformations, which we call the symmetric class, we define a new self-dual induction process in which the system is successively induced on a union of sub-intervals. This algorithm gives rise to an underlying graph structure which reflects the dynamical behavior of the system, through the Rokhlin towers of the induced maps. We apply it to build a wide assortment of explicit examples on four intervals having different dynamical properties: these include the first nontrivial examples with eigenvalues (rational or irrational), the first ever example of an exchange on more than three intervals satisfying Veech’s simplicity (though this weakening of the notion of minimal self-joinings was designed in 1982 to be satisfied by interval exchange transformations), and an unexpected example which is non uniquely ergodic, weakly mixing for one invariant ergodic measure but has rational eigenvalues for the other invariant ergodic measure.

LA - eng

KW - interval exchanges; self-dual induction; eigenvalues; Veech simplicity

UR - http://eudml.org/doc/272107

ER -

## References

top- [1] V. I. Arnolʼd, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk 18 (1963), 91–192, translation: Russian Math. Surveys 18 (1963), 86–194. Zbl0135.42701MR170705
- [2] P. Arnoux, Un exemple de semi-conjugaison entre un échange d’intervalles et une translation sur le tore, Bull. Soc. Math. France116 (1988), 489–500. Zbl0703.58045MR1005392
- [3] P. Arnoux & G. Rauzy, Représentation géométrique de suites de complexité $2n+1$, Bull. Soc. Math. France119 (1991), 199–215. Zbl0789.28011MR1116845
- [4] A. Avila & G. Forni, Weak mixing for interval exchange transformations and translation flows, Ann. of Math.165 (2007), 637–664. Zbl1136.37003MR2299743
- [5] M. Boshernitzan, A unique ergodicity of minimal symbolic flows with linear block growth, J. Anal. Math. 44 (1984/85), 77–96. Zbl0602.28008MR801288
- [6] M. Boshernitzan, A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J.52 (1985), 723–752. Zbl0602.28009MR808101
- [7] J. Cassaigne, S. Ferenczi & A. Messaoudi, Weak mixing and eigenvalues for Arnoux–Rauzy sequences, Ann. Inst. Fourier (Grenoble) 58 (2008), 1983–2005. Zbl1151.37013MR2473626
- [8] J. Chaika, There exists a topologically mixing IET, preprint arXiv:0910.3986.
- [9] J. R. Choksi & M. G. Nadkarni, The group of eigenvalues of a rank one transformation, Canad. Math. Bull.38 (1995), 42–54. Zbl0833.28008MR1319899
- [10] I. P. Cornfeld, S. V. Fomin & Y. G. Sinaĭ, Ergodic theory, Grund. Math. Wiss. 245, Springer, 1982. Zbl0493.28007MR832433
- [11] S. D. Cruz & L. F. C. da Rocha, A generalization of the Gauss map and some classical theorems on continued fractions, Nonlinearity18 (2005), 505–525. Zbl1153.11320MR2122671
- [12] S. Ferenczi, Systems of finite rank, Colloq. Math.73 (1997), 35–65. Zbl0883.28014MR1436950
- [13] S. Ferenczi, C. Holton & L. Q. Zamboni, Structure of three interval exchange transformations. I. An arithmetic study, Ann. Inst. Fourier (Grenoble) 51 (2001), 861–901. Zbl1029.11036MR1849209
- [14] S. Ferenczi, C. Holton & L. Q. Zamboni, Structure of three-interval exchange transformations. II. A combinatorial description of the trajectories, J. Anal. Math. 89 (2003), 239–276. Zbl1130.37324MR1981920
- [15] S. Ferenczi, C. Holton & L. Q. Zamboni, Structure of three-interval exchange transformations III. Ergodic and spectral properties, J. Anal. Math. 93 (2004), 103–138. Zbl1094.37005MR2110326
- [16] S. Ferenczi, C. Holton & L. Q. Zamboni, Joinings of three-interval exchange transformations, Ergodic Theory Dynam. Systems25 (2005), 483–502. Zbl1076.37002MR2129107
- [17] S. Ferenczi & C. Mauduit, Transcendence of numbers with a low complexity expansion, J. Number Theory67 (1997), 146–161. Zbl0895.11029MR1486494
- [18] S. Ferenczi & L. F. C. da Rocha, A self-dual induction for three-interval exchange transformations, Dyn. Syst.24 (2009), 393–412. Zbl1230.37005
- [19] S. Ferenczi & L. Q. Zamboni, Structure of $k$-interval exchange transformations: induction, trajectories, and distance theorems, J. Anal. Math.112 (2010), 289–328. Zbl1225.37003
- [20] H. Hmili, Non topologically weakly mixing interval exchanges, Discrete Contin. Dyn. Syst.27 (2010), 1079–1091. Zbl1195.37025
- [21] A. del Junco, A family of counterexamples in ergodic theory, Israel J. Math.44 (1983), 160–188. Zbl0522.28012
- [22] A. del Junco & D. J. Rudolph, A rank-one, rigid, simple, prime map, Ergodic Theory Dynam. Systems7 (1987), 229–247. Zbl0634.54020
- [23] A. B. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR211 (1973), 775–778. Zbl0298.28013
- [24] A. B. Katok & A. M. Stepin, Approximations in ergodic theory, Uspehi Mat. Nauk 22 (1967), 81–106, translation: Russian Math. Surveys 22 (1967), 76–102. Zbl0172.07202
- [25] M. Keane, Interval exchange transformations, Math. Z.141 (1975), 25–31. Zbl0278.28010
- [26] M. Keane, Non-ergodic interval exchange transformations, Israel J. Math.26 (1977), 188–196. Zbl0351.28012
- [27] A. O. Lopes & L. F. C. da Rocha, Invariant measures for Gauss maps associated with interval exchange maps, Indiana Univ. Math. J.43 (1994), 1399–1438. Zbl0840.28007
- [28] S. Marmi, P. Moussa & J.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps, J. Amer. Math. Soc.18 (2005), 823–872. Zbl1112.37002
- [29] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math.115 (1982), 169–200. Zbl0497.28012
- [30] A. Nogueira & D. J. Rudolph, Topological weak-mixing of interval exchange maps, Ergodic Theory Dynam. Systems17 (1997), 1183–1209. Zbl0958.37010
- [31] D. S. Ornstein, D. J. Rudolph & B. Weiss, Equivalence of measure preserving transformations, Mem. Amer. Math. Soc. 37 (1982). Zbl0504.28019
- [32] V. I. Oseledec, The spectrum of ergodic automorphisms, Dokl. Akad. Nauk SSSR168 (1966), 1009–1011. Zbl0152.33404
- [33] R. C. Penner & J. L. Harer, Combinatorics of train tracks, Annals of Math. Studies 125, Princeton Univ. Press, 1992. Zbl0765.57001MR1144770
- [34] G. Rauzy, Échanges d’intervalles et transformations induites, Acta Arith.34 (1979), 315–328. Zbl0414.28018
- [35] Y. G. Sinai & C. Ulcigrai, Weak mixing in interval exchange transformations of periodic type, Lett. Math. Phys.74 (2005), 111–133. Zbl1105.37002MR2191950
- [36] W. A. Veech, Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem $\mathrm{mod}\phantom{\rule{4pt}{0ex}}2$, Trans. Amer. Math. Soc.140 (1969), 1–33. Zbl0201.05601
- [37] W. A. Veech, Interval exchange transformations, J. Anal. Math.33 (1978), 222–272. Zbl0455.28006MR516048
- [38] W. A. Veech, A criterion for a process to be prime, Monatsh. Math.94 (1982), 335–341. Zbl0499.28016MR685378
- [39] W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math.115 (1982), 201–242. Zbl0486.28014
- [40] W. A. Veech, The metric theory of interval exchange transformations. I. Generic spectral properties, Amer. J. Math. 106 (1984), 1331–1359. Zbl0631.28006MR765582
- [41] M. Viana, Dynamics of interval exchange maps and Teichmüller flows, preprint http://w3.impa.br/~viana/out/ietf.pdf.
- [42] J.-C. Yoccoz, Échanges d’intervalles, cours au Collège de France, http://www.college-de-france.fr/media/equ_dif/UPL8726_yoccoz05.pdf, 2005.
- [43] J.-C. Yoccoz, Continued fraction algorithms for interval exchange maps: an introduction, in Frontiers in number theory, physics, and geometry. I, Springer, 2006, 401–435. Zbl1127.28011MR2261103
- [44] A. Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble) 46 (1996), 325–370. Zbl0853.28007MR1393518

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