# American Institute of Mathematical Sciences

2017, 11: 219-248. doi: 10.3934/jmd.2017010

## Minimality of interval exchange transformations with restrictions

 Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Str., Moscow 119991, Russia

Received  October 12, 2015 Revised  December 12, 2016 Published  March 2017

Fund Project: The work is supported by the Russian Science Foundation under grant 14-50-00005 and performed at Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

It is known since a 40-year-old paper by M.Keane that minimality is a generic (i.e., holding with probability one) property of an irreducible interval exchange transformation. If one puts some integral linear restrictions on the parameters of the interval exchange transformation, then minimality may become an "exotic" property. We conjecture in this paper that this occurs if and only if the linear restrictions contain a Lagrangian subspace of the first homology of the suspension surface. We partially prove it in the `only if' direction and provide a series of examples to support the converse one. We show that the unique ergodicity remains a generic property if the restrictions on the parameters do not contain a Lagrangian subspace (this result is due to Barak Weiss).

Citation: Ivan Dynnikov, Alexandra Skripchenko. Minimality of interval exchange transformations with restrictions. Journal of Modern Dynamics, 2017, 11: 219-248. doi: 10.3934/jmd.2017010
##### References:
 [1] V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, (Russian) Uspekhi Mat. Nauk, 18 (1963), 91-192; translation in Russian Math. Surveys, 18 (1963), 85-191. Google Scholar [2] P. Arnoux, Échanges d'intervalles et flots sur les surfaces, (French) in Ergodic Theory (Sem., Les Plans-sur-Bex, 1980), Monograph. Enseign. Math., 29, Univ. Genéve, Geneva, 1981, 5-38. Google Scholar [3] P. Arnoux and T. Schmidt, Veech surfaces with nonperiodic directions in the trace field, J. Mod. Dyn., 3 (2009), 611-629. doi: 10.3934/jmd.2009.3.611. Google Scholar [4] P. Arnoux and J.-C. Yoccoz, Construction de difféomorphismes pseudo-Anosov, (French), C. R. Acad. Sc. Paris Sr. I Math., 292 (1981), 75-78. Google Scholar [5] A. Avila, P. Hubert and A. Skripchenko, On the Hausdorff dimension of the Rauzy gasket, Bul. Soc. Math. France, 144 (2016), 539-568. doi: 10.24033/bsmf.2722. Google Scholar [6] P. Arnoux and Š. Starosta, Rauzy gasket, in Further Developments in Fractals and Related Fields: Mathematical Foundations and Connections, Trends Math., Birkhäuser/Springer, New York, 2013, 1-23. doi: 10.1007/978-0-8176-8400-6_1. Google Scholar [7] M. Bestvina and M. Feign, Stable actions of groups on real trees, Invent. Math., 121 (1995), 287-321. doi: 10.1007/BF01884300. Google Scholar [8] M. Boshernitzan, Rank two interval exchange transformations, Ergodic Theory Dynam. Systems, 8 (1988), 379-394. doi: 10.1017/S0143385700004521. Google Scholar [9] H. Bruin and S. Troubetzkoy, The Gauss map on a class of interval translation mappings, Israel J. Math., 137 (2003), 125-148. doi: 10.1007/BF02785958. Google Scholar [10] M. Boshernitzan and I. Kornfeld, Interval translation mappings, Ergodic Theory Dynam. Systems, 15 (1995), 821-832. doi: 10.1017/S0143385700009652. Google Scholar [11] H. Bruin and G. Clack, Inducing and unique ergodicity of double rotations, Discrete Contin. Dyn. Syst., 32 (2012), 4133-4147. doi: 10.3934/dcds.2012.32.4133. Google Scholar [12] R. De Leo and I. Dynnikov, Geometry of plane sections of the infinite regular skew polyhedron {4, 6|4}, Geom. Dedicata, 138 (2009), 51-67. doi: 10.1007/s10711-008-9298-1. Google Scholar [13] I. Dynnikov, A proof of Novikov's conjecture on semiclassical motion of an electron, (Russian) Mat. Zametki, 53 (1993), 57-68; translation in Math. Notes, 53 (1993), 495-501. doi: 10.1007/BF01208544. Google Scholar [14] I. Dynnikov, Interval identification systems and plane sections of 3-periodic surfaces, (Russian) Tr. Mat. Inst. Steklova, 263 (2008), Geometriya, Topologiya i Matematicheskaya Fizika. I, 72-84; translation in Proc. Steklov Inst. Math. , 263 (2008), 65-77. doi: 10.1134/S008154380. Google Scholar [15] I. Dynnikov and A. Skripchenko, On typical leaves of a measured foliated 2-complex of thin type, in Topology, Geometry, Integrable Systems, and Mathematical Physics: Novikov's Seminar 2012-2014 (eds. V. M. Buchstaber, B. A. Dubrovin and I. M. Krichever), Amer. Math. Soc. Transl. Ser. 2,234, Amer. Math. Soc., Providence, RI, 2014,173-200.Google Scholar [16] C. Danthony and A. Nogueira, Measured foliations on nonorientable surfaces, Ann. Sci. école. Norm. Sup., 23 (1990), 469-494. Google Scholar [17] D. Gaboriau, G. Levitt and F. Paulin, Pseudogroups of isometries of R and Rips' theorem on free actions on R-trees, Israel J. Math., 87 (1994), 403-428. doi: 10.1007/BF02773004. Google Scholar [18] H. Imanishi, On codimension one foliations defined by closed one forms with singularities, J. Math. Kyoto Univ., 19 (1979), 285-291. doi: 10.1215/kjm/1250522432. Google Scholar [19] A. Katok and A. Stepin, Approximations in ergodic theory, (Russian) Uspekhi Mat. Nauk, 22 (1967), 81-106; translation in Russian Math. Surveys, 22 (1967), 77-102. Google Scholar [20] M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981. Google Scholar [21] M. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. doi: 10.1007/BF03007668. Google Scholar [22] R. Kenyon and J. Smillie, Billiards on rational-angled triangles, Comment. Math. Helv., 75 (2000), 65-108. doi: 10.1007/s000140050113. Google Scholar [23] S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. Ser. 2, 124 (1986), 293-311. doi: 10.2307/1971280. Google Scholar [24] G. Levitt, The dynamics of rotation pseudogroups, (French), Invent. Math., 113 (1993), 633-670. doi: 10.1007/BF01244321. Google Scholar [25] A. Ya. Maltsev and S. P. Novikov, Dynamical systems, topology, and conductivity in normal metals, J. Statist. Phys., 115 (2004), 31-46. doi: 10.1023/B:JOSS.0000. Google Scholar [26] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math., 115 (1982), 169-200. doi: 10.2307/1971341. Google Scholar [27] H. Masur, Ergodic theory of translation surfaces, in Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006,527-547. doi: 10.1016/S1874-575X(06)80032-9. Google Scholar [28] C. McMullen, Teichmüller geodesics of infinite complexity, Acta Math., 191 (2003), 191-223. doi: 10.1007/BF02392964. Google Scholar [29] Y. Minsky and B. Weiss, Weiss, Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges, Ann. Sci. Éc. Norm. Supér., 47 (2014), 245-284. Google Scholar [30] S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, (Russian), Uspekhi Mat. Nauk, 37 (1982), 3-49. Google Scholar [31] Ch. Novak, Group actions via interval exchange transformations, PhD Thesis, Northwestern University, 2008. Google Scholar [32] A. Nogueira, Almost all interval exchange transformations with flips are nonergodic, Ergodic Theory Dynam. Systems, 9 (1989), 515-525. doi: 10.1017/S0143385700005150. Google Scholar [33] V. I. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, (Russian) Trudy Moskov. Mat. ObŠč., 19 (1968), 179-210; translation in Trans. Moscow Math. Soc. , 19 (1968), 197-231. Google Scholar [34] G. Rauzy, échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328. Google Scholar [35] H. Suzuki, S. Ito and K. Aihara, Double rotations, Discrete Contin. Dyn. Syst., 13 (2005), 515-532. doi: 10.3934/dcds.2005.13.515. Google Scholar [36] S. Schwartzman, Asymptotic cycles, Annals of Mathematics, Second Series, 66 (1957), 270-284. doi: 10.2307/1969999. Google Scholar [37] W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. Math., 115 (1982), 201-242. doi: 10.2307/2374398. Google Scholar [38] W. Veech, The metric theory of interval exchange transformations Ⅲ, The Sah-Arnoux-Fathi, American Journal of Mathematics, 106 (1984), 1389-1422. doi: 10.2307/1971391. Google Scholar [39] M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100. doi: 10.5209/rev_REMA.2006.v19.n1.16621. Google Scholar [40] M. Viana, Lyapunov exponents of Teichmüller flows, Partially hyperbolic dynamics, laminations and Teichmüller flows, eds. G. Forni, M. Lyubich, Ch. Pugh, and M. Shub, 51 (2007), 139-201. Google Scholar [41] D. Volk, Almost every interval translation map of three intervals is finite type, Discrete Contin. Dyn. Syst., 34 (2014), 2307-2314. doi: 10.3934/dcds.2014.34.2307. Google Scholar [42] A. Zorich, A problem of Novikov on the semiclassical motion of an electron in a uniform almost rational magnetic field, (Russian) Uspekhi Mat. Nauk, 39 (1984), 235-236; translation in Russian Math. Surveys, 39 (1984), 287-288. Google Scholar [43] A. Zorich, Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477-1499. doi: 10.1017/S0143385797086215. Google Scholar [44] A. Zorich, How do the leaves of a closed 1-form wind around a surface?, in Pseudoperiodic Topology (eds. V. Arnold, M. Kontsevich and A. Zorich), Amer. Math. Soc. Transl. Ser. 2,197, Amer. Math. Soc., Providence, RI, 1999,135-178. doi: 10.1090/trans2/197. Google Scholar [45] A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry I (eds. P. Cartier, B. Julia and P. Moussa), Springer Verlag, 2006,437-583. doi: 10.1007/978-3-540-31347-2_13. Google Scholar

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##### References:
 [1] V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, (Russian) Uspekhi Mat. Nauk, 18 (1963), 91-192; translation in Russian Math. Surveys, 18 (1963), 85-191. Google Scholar [2] P. Arnoux, Échanges d'intervalles et flots sur les surfaces, (French) in Ergodic Theory (Sem., Les Plans-sur-Bex, 1980), Monograph. Enseign. Math., 29, Univ. Genéve, Geneva, 1981, 5-38. Google Scholar [3] P. Arnoux and T. Schmidt, Veech surfaces with nonperiodic directions in the trace field, J. Mod. Dyn., 3 (2009), 611-629. doi: 10.3934/jmd.2009.3.611. Google Scholar [4] P. Arnoux and J.-C. Yoccoz, Construction de difféomorphismes pseudo-Anosov, (French), C. R. Acad. Sc. Paris Sr. I Math., 292 (1981), 75-78. Google Scholar [5] A. Avila, P. Hubert and A. Skripchenko, On the Hausdorff dimension of the Rauzy gasket, Bul. Soc. Math. France, 144 (2016), 539-568. doi: 10.24033/bsmf.2722. Google Scholar [6] P. Arnoux and Š. Starosta, Rauzy gasket, in Further Developments in Fractals and Related Fields: Mathematical Foundations and Connections, Trends Math., Birkhäuser/Springer, New York, 2013, 1-23. doi: 10.1007/978-0-8176-8400-6_1. Google Scholar [7] M. Bestvina and M. Feign, Stable actions of groups on real trees, Invent. Math., 121 (1995), 287-321. doi: 10.1007/BF01884300. Google Scholar [8] M. Boshernitzan, Rank two interval exchange transformations, Ergodic Theory Dynam. Systems, 8 (1988), 379-394. doi: 10.1017/S0143385700004521. Google Scholar [9] H. Bruin and S. Troubetzkoy, The Gauss map on a class of interval translation mappings, Israel J. Math., 137 (2003), 125-148. doi: 10.1007/BF02785958. Google Scholar [10] M. Boshernitzan and I. Kornfeld, Interval translation mappings, Ergodic Theory Dynam. Systems, 15 (1995), 821-832. doi: 10.1017/S0143385700009652. Google Scholar [11] H. Bruin and G. Clack, Inducing and unique ergodicity of double rotations, Discrete Contin. Dyn. Syst., 32 (2012), 4133-4147. doi: 10.3934/dcds.2012.32.4133. Google Scholar [12] R. De Leo and I. Dynnikov, Geometry of plane sections of the infinite regular skew polyhedron {4, 6|4}, Geom. Dedicata, 138 (2009), 51-67. doi: 10.1007/s10711-008-9298-1. Google Scholar [13] I. Dynnikov, A proof of Novikov's conjecture on semiclassical motion of an electron, (Russian) Mat. Zametki, 53 (1993), 57-68; translation in Math. Notes, 53 (1993), 495-501. doi: 10.1007/BF01208544. Google Scholar [14] I. Dynnikov, Interval identification systems and plane sections of 3-periodic surfaces, (Russian) Tr. Mat. Inst. Steklova, 263 (2008), Geometriya, Topologiya i Matematicheskaya Fizika. I, 72-84; translation in Proc. Steklov Inst. Math. , 263 (2008), 65-77. doi: 10.1134/S008154380. Google Scholar [15] I. Dynnikov and A. Skripchenko, On typical leaves of a measured foliated 2-complex of thin type, in Topology, Geometry, Integrable Systems, and Mathematical Physics: Novikov's Seminar 2012-2014 (eds. V. M. Buchstaber, B. A. Dubrovin and I. M. Krichever), Amer. Math. Soc. Transl. Ser. 2,234, Amer. Math. Soc., Providence, RI, 2014,173-200.Google Scholar [16] C. Danthony and A. Nogueira, Measured foliations on nonorientable surfaces, Ann. Sci. école. Norm. Sup., 23 (1990), 469-494. Google Scholar [17] D. Gaboriau, G. Levitt and F. Paulin, Pseudogroups of isometries of R and Rips' theorem on free actions on R-trees, Israel J. Math., 87 (1994), 403-428. doi: 10.1007/BF02773004. Google Scholar [18] H. Imanishi, On codimension one foliations defined by closed one forms with singularities, J. Math. Kyoto Univ., 19 (1979), 285-291. doi: 10.1215/kjm/1250522432. Google Scholar [19] A. Katok and A. Stepin, Approximations in ergodic theory, (Russian) Uspekhi Mat. Nauk, 22 (1967), 81-106; translation in Russian Math. Surveys, 22 (1967), 77-102. Google Scholar [20] M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981. Google Scholar [21] M. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. doi: 10.1007/BF03007668. Google Scholar [22] R. Kenyon and J. Smillie, Billiards on rational-angled triangles, Comment. Math. Helv., 75 (2000), 65-108. doi: 10.1007/s000140050113. Google Scholar [23] S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. Ser. 2, 124 (1986), 293-311. doi: 10.2307/1971280. Google Scholar [24] G. Levitt, The dynamics of rotation pseudogroups, (French), Invent. Math., 113 (1993), 633-670. doi: 10.1007/BF01244321. Google Scholar [25] A. Ya. Maltsev and S. P. Novikov, Dynamical systems, topology, and conductivity in normal metals, J. Statist. Phys., 115 (2004), 31-46. doi: 10.1023/B:JOSS.0000. Google Scholar [26] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math., 115 (1982), 169-200. doi: 10.2307/1971341. Google Scholar [27] H. Masur, Ergodic theory of translation surfaces, in Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006,527-547. doi: 10.1016/S1874-575X(06)80032-9. Google Scholar [28] C. McMullen, Teichmüller geodesics of infinite complexity, Acta Math., 191 (2003), 191-223. doi: 10.1007/BF02392964. Google Scholar [29] Y. Minsky and B. Weiss, Weiss, Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges, Ann. Sci. Éc. Norm. Supér., 47 (2014), 245-284. Google Scholar [30] S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, (Russian), Uspekhi Mat. Nauk, 37 (1982), 3-49. Google Scholar [31] Ch. Novak, Group actions via interval exchange transformations, PhD Thesis, Northwestern University, 2008. Google Scholar [32] A. Nogueira, Almost all interval exchange transformations with flips are nonergodic, Ergodic Theory Dynam. Systems, 9 (1989), 515-525. doi: 10.1017/S0143385700005150. Google Scholar [33] V. I. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, (Russian) Trudy Moskov. Mat. ObŠč., 19 (1968), 179-210; translation in Trans. Moscow Math. Soc. , 19 (1968), 197-231. Google Scholar [34] G. Rauzy, échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328. Google Scholar [35] H. Suzuki, S. Ito and K. Aihara, Double rotations, Discrete Contin. Dyn. Syst., 13 (2005), 515-532. doi: 10.3934/dcds.2005.13.515. Google Scholar [36] S. Schwartzman, Asymptotic cycles, Annals of Mathematics, Second Series, 66 (1957), 270-284. doi: 10.2307/1969999. Google Scholar [37] W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. Math., 115 (1982), 201-242. doi: 10.2307/2374398. Google Scholar [38] W. Veech, The metric theory of interval exchange transformations Ⅲ, The Sah-Arnoux-Fathi, American Journal of Mathematics, 106 (1984), 1389-1422. doi: 10.2307/1971391. Google Scholar [39] M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100. doi: 10.5209/rev_REMA.2006.v19.n1.16621. Google Scholar [40] M. Viana, Lyapunov exponents of Teichmüller flows, Partially hyperbolic dynamics, laminations and Teichmüller flows, eds. G. Forni, M. Lyubich, Ch. Pugh, and M. Shub, 51 (2007), 139-201. Google Scholar [41] D. Volk, Almost every interval translation map of three intervals is finite type, Discrete Contin. Dyn. Syst., 34 (2014), 2307-2314. doi: 10.3934/dcds.2014.34.2307. Google Scholar [42] A. Zorich, A problem of Novikov on the semiclassical motion of an electron in a uniform almost rational magnetic field, (Russian) Uspekhi Mat. Nauk, 39 (1984), 235-236; translation in Russian Math. Surveys, 39 (1984), 287-288. Google Scholar [43] A. Zorich, Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477-1499. doi: 10.1017/S0143385797086215. Google Scholar [44] A. Zorich, How do the leaves of a closed 1-form wind around a surface?, in Pseudoperiodic Topology (eds. V. Arnold, M. Kontsevich and A. Zorich), Amer. Math. Soc. Transl. Ser. 2,197, Amer. Math. Soc., Providence, RI, 1999,135-178. doi: 10.1090/trans2/197. Google Scholar [45] A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry I (eds. P. Cartier, B. Julia and P. Moussa), Springer Verlag, 2006,437-583. doi: 10.1007/978-3-540-31347-2_13. Google Scholar
The subset $M(\pi, \mathscr U)\subset\Delta^n\cap\mathscr U$ in Examples 1.7 (left) and 1.8 (right). Only points with $\dim_{\mathbb Q}\langle a_1, a_2, 1\rangle=3$ are considered
Singularities of $\mathscr F_{\pi, \mathbf a}$
A transversal representing a separating cycle (bold line) and the restriction cycle (dashed line) in Example 2.18
Double suspension surface. Each vertical straight line segment in $\partial D$ is collapsed to a point, the bottom side of the square is identified with the top one
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