2019, 14: 153-177. doi: 10.3934/jmd.2019006

Rigidity of square-tiled interval exchange transformations

Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, I2M - UMR 7373, 13453 Marseille, France

To the memory of William Veech whose mathematics were a constant source of inspiration for both authors, and who always showed great kindness to the members of the Marseille school, beginning with its founder Gérard Rauzy

Received  March 02, 2017 Revised  April 24, 2018 Published  March 2019

We look at interval exchange transformations defined as first return maps on the set of diagonals of a flow of direction $ \theta $ on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if $ \tan\theta $ has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and $ \tan\theta $ has bounded partial quotients, the square-tiled interval exchange transformation $ T $ is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an uncountable set of rigid directional flows and an uncountable set of rigid interval exchange transformations.

Citation: Sébastien Ferenczi, Pascal Hubert. Rigidity of square-tiled interval exchange transformations. Journal of Modern Dynamics, 2019, 14: 153-177. doi: 10.3934/jmd.2019006
References:
[1]

O. N. Ageev, The spectral multiplicity function and geometric representations of interval exchange transformations, Sb. Math., 190 (1999), 1-28. doi: 10.1070/SM1999v190n01ABEH000376.

[2]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Math. Surveys, 18 (1963), 86-194.

[3]

P. Arnoux, Un exemple de semi-conjugaison entre un échange d'intervalles et une translation sur le tore, Bull. Soc. Math. France, 116 (1988), 489-500. doi: 10.24033/bsmf.2109.

[4]

P. ArnouxJ. Bernat and X. Bressaud, Geometrical models for substitutions, Exp. Math., 20 (2011), 97-127. doi: 10.1080/10586458.2011.544590.

[5]

P. Arnoux and J. C. Yoccoz, Construction de difféomorphismes pseudo-Anosov, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 75-78.

[6]

A. Avila and G. Forni, Weak mixing for interval exchange maps and translation flows, Ann. of Math. (2), 165 (2007), 637-664. doi: 10.4007/annals.2007.165.637.

[7]

A. Avila and V. Delecroix, Weak mixing directions in non-arithmetic Veech surfaces, J. Amer. Math. Soc., 29 (2016), 1167-1208. doi: 10.1090/jams/856.

[8]

V. BerthéN. Chekhova and S. Ferenczi, Covering numbers: Arithmetics and dynamics for rotations and interval exchanges, J. Anal. Math., 79 (1999), 1-31. doi: 10.1007/BF02788235.

[9]

M. Boshernitzan, Rank two interval exchange transformations, Ergodic Theory Dynam. Systems, 8 (1988), 379-394. doi: 10.1017/S0143385700004521.

[10]

N. Chekhova, Covering numbers of rotations, Theoret. Comput. Sci., 230 (2000), 97-116. doi: 10.1016/S0304-3975(97)00256-9.

[11]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL(2, $ {\mathbb R}$) action on moduli space, Publ. Math. Inst. Hautes Études Sci., 127 (2018), 95–324. doi: 10.1007/s10240-018-0099-2.

[12]

A. EskinM. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the SL(2, $ {\mathbb R}$) action on moduli space, Ann. of Math. (2), 182 (2015), 673-721. doi: 10.4007/annals.2015.182.2.7.

[13]

S. Ferenczi, Measure-theoretic complexity of ergodic systems, Israel J. Math., 100 (1997), 189-207. doi: 10.1007/BF02773640.

[14]

S. Ferenczi, Systems of finite rank, Colloq. Math., 73 (1997), 35-65. doi: 10.4064/cm-73-1-35-65.

[15]

S. Ferenczi, Billiards in regular $2n$-gons and the self-dual induction, J. Lond. Math. Soc. (2), 87 (2013), 766-784. doi: 10.1112/jlms/jds075.

[16]

S. Ferenczi, A generalization of the self-dual induction to every interval exchange transformation, Ann. Inst. Fourier (Grenoble), 64 (2014), 1947-2002. doi: 10.5802/aif.2901.

[17]

S. FerencziC. Holton and L. Q. Zamboni, Joinings of three-interval exchange transformations, Ergodic Th. Dyn. Syst., 25 (2005), 483-502. doi: 10.1017/S0143385704000811.

[18]

S. Ferenczi and L. Q. Zamboni, Structure of K-interval exchange transformations: Induction, trajectories, and distance theorems, J. Anal. Math., 112 (2010), 289-328. doi: 10.1007/s11854-010-0031-2.

[19]

S. Ferenczi and L. Q. Zamboni, Eigenvalues and simplicity for interval exchange transformations, Ann. Sci. Ec. Norm. Sup., 4, 44 (2011), 361-392. doi: 10.24033/asens.2145.

[20]

K. Frączek, Diversity of mild mixing property for vertical flows of abelian differentials, Proc. Amer. Math. Soc., 137 (2009), 4229-4142. doi: 10.1090/S0002-9939-09-10025-4.

[21]

H. Hmili, Non topologically weakly mixing interval exchanges, Discrete Contin. Dyn. Syst., 27 (2010), 1079-1091. doi: 10.3934/dcds.2010.27.1079.

[22]

A. del Junco, A transformation with simple spectrum which is not rank one, Canad. J. Math., 29 (1977), 655-663. doi: 10.4153/CJM-1977-067-7.

[23]

A. Kanigowski and M. Lemańczyk, Flows with Ratner's property have discrete essential centralizer, Studia Math., 237 (2017), 185-194. doi: 10.4064/sm8660-11-2016.

[24]

A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Math. Surveys, 22 (1967), 76-102.

[25]

M. S. Keane, Interval exchange transformations, Math. Zeitsch., 141 (1975), 25-31. doi: 10.1007/BF01236981.

[26]

S. KerckhoffH. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2), 124 (1986), 293-311. doi: 10.2307/1971280.

[27]

M. Lemańczyk and M. K. Mentzen, On metric properties of substitutions, Compositio Math., 65 (1988), 241-263.

[28]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341.

[29]

S. Munday, On Hausdorff dimension and cusp excursions for Fuchsian groups, Discrete Contin. Dyn. Syst., 32 (2012), 2503-2520. doi: 10.3934/dcds.2012.32.2503.

[30]

V. I. Oseledec, The spectrum of ergodic automorphisms, Soviet Math. Doklady, 7 (1966), 776-779.

[31]

G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328. doi: 10.4064/aa-34-4-315-328.

[32]

D. Robertson, Mild mixing of certain interval exchange transformations, Ergodic Theory Dynam. Systems, 39 (2019), 248-256. doi: 10.1017/etds.2017.31.

[33]

E. A. Robinson and Jr ., Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math., 72 (1983), 299-314. doi: 10.1007/BF01389325.

[34]

P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, 2011, http://publications.ias.edu/sarnak/paper/512.

[35]

J. Smillie and C. Ulcigrai, Beyond Sturmian sequences: Coding linear trajectories in the regular octagon, Proc. Lond. Math. Soc. (3), 102 (2011), 291-340. doi: 10.1112/plms/pdq018.

[36]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.

[37]

W. A. Veech, A criterion for a process to be prime, Monatsh. Math., 94 (1982), 335-341. doi: 10.1007/BF01667386.

[38]

W. A. Veech, The metric theory of interval exchange transformations. I. Generic spectral properties, Amer. J. Math., 106 (1984), 1331-1359. doi: 10.2307/2374396.

[39]

W. A. Veech, A. Boshernitzan's criterion for unique ergodicity of an interval exchange transformation, Ergodic Theory Dynam. Systems, 7 (1987), 149-153. doi: 10.1017/S0143385700003862.

[40]

W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.

[41]

M. Viana, Dynamics of interval exchange maps and Teichmüller flows, preliminary manuscript, http://w3.impa.br/~viana/out/ietf.pdf.

[42]

A. Wright, Translation surfaces and their orbit closures: An introduction for a broad audience, EMS Surv. Math. Sci., 2 (2015), 63-108. doi: 10.4171/EMSS/9.

[43]

K. Yancey, Dynamics of self-similar interval exchange transformations on three intervals, in Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby, Contemp. Math., 678, Amer. Math. Soc., Providence, RI, 2016, 297–316.

[44]

J.-C. Yoccoz, Échanges d'intervalles (in French), Cours au Collège de France, 2005, http://www.college-de-france.fr/site/jean-christophe-yoccoz/.

[45]

D. Zmiaikou, Origami et Groupes de Permutation, Ph. D. thesis, http://www.zmiaikou.com/research.

[46]

A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006, 437–583. doi: 10.1007/978-3-540-31347-2_13.

show all references

References:
[1]

O. N. Ageev, The spectral multiplicity function and geometric representations of interval exchange transformations, Sb. Math., 190 (1999), 1-28. doi: 10.1070/SM1999v190n01ABEH000376.

[2]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Math. Surveys, 18 (1963), 86-194.

[3]

P. Arnoux, Un exemple de semi-conjugaison entre un échange d'intervalles et une translation sur le tore, Bull. Soc. Math. France, 116 (1988), 489-500. doi: 10.24033/bsmf.2109.

[4]

P. ArnouxJ. Bernat and X. Bressaud, Geometrical models for substitutions, Exp. Math., 20 (2011), 97-127. doi: 10.1080/10586458.2011.544590.

[5]

P. Arnoux and J. C. Yoccoz, Construction de difféomorphismes pseudo-Anosov, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 75-78.

[6]

A. Avila and G. Forni, Weak mixing for interval exchange maps and translation flows, Ann. of Math. (2), 165 (2007), 637-664. doi: 10.4007/annals.2007.165.637.

[7]

A. Avila and V. Delecroix, Weak mixing directions in non-arithmetic Veech surfaces, J. Amer. Math. Soc., 29 (2016), 1167-1208. doi: 10.1090/jams/856.

[8]

V. BerthéN. Chekhova and S. Ferenczi, Covering numbers: Arithmetics and dynamics for rotations and interval exchanges, J. Anal. Math., 79 (1999), 1-31. doi: 10.1007/BF02788235.

[9]

M. Boshernitzan, Rank two interval exchange transformations, Ergodic Theory Dynam. Systems, 8 (1988), 379-394. doi: 10.1017/S0143385700004521.

[10]

N. Chekhova, Covering numbers of rotations, Theoret. Comput. Sci., 230 (2000), 97-116. doi: 10.1016/S0304-3975(97)00256-9.

[11]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL(2, $ {\mathbb R}$) action on moduli space, Publ. Math. Inst. Hautes Études Sci., 127 (2018), 95–324. doi: 10.1007/s10240-018-0099-2.

[12]

A. EskinM. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the SL(2, $ {\mathbb R}$) action on moduli space, Ann. of Math. (2), 182 (2015), 673-721. doi: 10.4007/annals.2015.182.2.7.

[13]

S. Ferenczi, Measure-theoretic complexity of ergodic systems, Israel J. Math., 100 (1997), 189-207. doi: 10.1007/BF02773640.

[14]

S. Ferenczi, Systems of finite rank, Colloq. Math., 73 (1997), 35-65. doi: 10.4064/cm-73-1-35-65.

[15]

S. Ferenczi, Billiards in regular $2n$-gons and the self-dual induction, J. Lond. Math. Soc. (2), 87 (2013), 766-784. doi: 10.1112/jlms/jds075.

[16]

S. Ferenczi, A generalization of the self-dual induction to every interval exchange transformation, Ann. Inst. Fourier (Grenoble), 64 (2014), 1947-2002. doi: 10.5802/aif.2901.

[17]

S. FerencziC. Holton and L. Q. Zamboni, Joinings of three-interval exchange transformations, Ergodic Th. Dyn. Syst., 25 (2005), 483-502. doi: 10.1017/S0143385704000811.

[18]

S. Ferenczi and L. Q. Zamboni, Structure of K-interval exchange transformations: Induction, trajectories, and distance theorems, J. Anal. Math., 112 (2010), 289-328. doi: 10.1007/s11854-010-0031-2.

[19]

S. Ferenczi and L. Q. Zamboni, Eigenvalues and simplicity for interval exchange transformations, Ann. Sci. Ec. Norm. Sup., 4, 44 (2011), 361-392. doi: 10.24033/asens.2145.

[20]

K. Frączek, Diversity of mild mixing property for vertical flows of abelian differentials, Proc. Amer. Math. Soc., 137 (2009), 4229-4142. doi: 10.1090/S0002-9939-09-10025-4.

[21]

H. Hmili, Non topologically weakly mixing interval exchanges, Discrete Contin. Dyn. Syst., 27 (2010), 1079-1091. doi: 10.3934/dcds.2010.27.1079.

[22]

A. del Junco, A transformation with simple spectrum which is not rank one, Canad. J. Math., 29 (1977), 655-663. doi: 10.4153/CJM-1977-067-7.

[23]

A. Kanigowski and M. Lemańczyk, Flows with Ratner's property have discrete essential centralizer, Studia Math., 237 (2017), 185-194. doi: 10.4064/sm8660-11-2016.

[24]

A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Math. Surveys, 22 (1967), 76-102.

[25]

M. S. Keane, Interval exchange transformations, Math. Zeitsch., 141 (1975), 25-31. doi: 10.1007/BF01236981.

[26]

S. KerckhoffH. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2), 124 (1986), 293-311. doi: 10.2307/1971280.

[27]

M. Lemańczyk and M. K. Mentzen, On metric properties of substitutions, Compositio Math., 65 (1988), 241-263.

[28]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341.

[29]

S. Munday, On Hausdorff dimension and cusp excursions for Fuchsian groups, Discrete Contin. Dyn. Syst., 32 (2012), 2503-2520. doi: 10.3934/dcds.2012.32.2503.

[30]

V. I. Oseledec, The spectrum of ergodic automorphisms, Soviet Math. Doklady, 7 (1966), 776-779.

[31]

G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328. doi: 10.4064/aa-34-4-315-328.

[32]

D. Robertson, Mild mixing of certain interval exchange transformations, Ergodic Theory Dynam. Systems, 39 (2019), 248-256. doi: 10.1017/etds.2017.31.

[33]

E. A. Robinson and Jr ., Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math., 72 (1983), 299-314. doi: 10.1007/BF01389325.

[34]

P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, 2011, http://publications.ias.edu/sarnak/paper/512.

[35]

J. Smillie and C. Ulcigrai, Beyond Sturmian sequences: Coding linear trajectories in the regular octagon, Proc. Lond. Math. Soc. (3), 102 (2011), 291-340. doi: 10.1112/plms/pdq018.

[36]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.

[37]

W. A. Veech, A criterion for a process to be prime, Monatsh. Math., 94 (1982), 335-341. doi: 10.1007/BF01667386.

[38]

W. A. Veech, The metric theory of interval exchange transformations. I. Generic spectral properties, Amer. J. Math., 106 (1984), 1331-1359. doi: 10.2307/2374396.

[39]

W. A. Veech, A. Boshernitzan's criterion for unique ergodicity of an interval exchange transformation, Ergodic Theory Dynam. Systems, 7 (1987), 149-153. doi: 10.1017/S0143385700003862.

[40]

W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.

[41]

M. Viana, Dynamics of interval exchange maps and Teichmüller flows, preliminary manuscript, http://w3.impa.br/~viana/out/ietf.pdf.

[42]

A. Wright, Translation surfaces and their orbit closures: An introduction for a broad audience, EMS Surv. Math. Sci., 2 (2015), 63-108. doi: 10.4171/EMSS/9.

[43]

K. Yancey, Dynamics of self-similar interval exchange transformations on three intervals, in Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby, Contemp. Math., 678, Amer. Math. Soc., Providence, RI, 2016, 297–316.

[44]

J.-C. Yoccoz, Échanges d'intervalles (in French), Cours au Collège de France, 2005, http://www.college-de-france.fr/site/jean-christophe-yoccoz/.

[45]

D. Zmiaikou, Origami et Groupes de Permutation, Ph. D. thesis, http://www.zmiaikou.com/research.

[46]

A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006, 437–583. doi: 10.1007/978-3-540-31347-2_13.

Figure 1.  Square-tiled surface of Example 30. Letters $ a, b, c, d $ describe the sides identifications
Figure 2.  Building an interval exchange associated to the surface in Example 30
Figure 3.  The square-tiled interval exchange we get in Example 30
Figure 4.  Regular octagon
Figure 5.  Interval exchange transformation in the regular octagon
Figure 6.  Trajectories in direction θ run along the cylinder from J in direction θn once, unless they are in the subinterval B
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