August 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.

[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.

[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.

[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.

[5]

A. AvilaP. 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.

[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.

[7]

M. Bestvina and M. Feign, Stable actions of groups on real trees, Invent. Math., 121 (1995), 287-321. doi: 10.1007/BF01884300.

[8]

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

[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.

[10]

M. Boshernitzan and I. Kornfeld, Interval translation mappings, Ergodic Theory Dynam. Systems, 15 (1995), 821-832. doi: 10.1017/S0143385700009652.

[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.

[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.

[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.

[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.

[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.

[16]

C. Danthony and A. Nogueira, Measured foliations on nonorientable surfaces, Ann. Sci. école. Norm. Sup., 23 (1990), 469-494.

[17]

D. GaboriauG. 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.

[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.

[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.

[20]

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

[21]

M. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. doi: 10.1007/BF03007668.

[22]

R. Kenyon and J. Smillie, Billiards on rational-angled triangles, Comment. Math. Helv., 75 (2000), 65-108. doi: 10.1007/s000140050113.

[23]

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

[24]

G. Levitt, The dynamics of rotation pseudogroups, (French), Invent. Math., 113 (1993), 633-670. doi: 10.1007/BF01244321.

[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.

[26]

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

[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.

[28]

C. McMullen, Teichmüller geodesics of infinite complexity, Acta Math., 191 (2003), 191-223. doi: 10.1007/BF02392964.

[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.

[30]

S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, (Russian), Uspekhi Mat. Nauk, 37 (1982), 3-49.

[31]

Ch. Novak, Group actions via interval exchange transformations, PhD Thesis, Northwestern University, 2008.

[32]

A. Nogueira, Almost all interval exchange transformations with flips are nonergodic, Ergodic Theory Dynam. Systems, 9 (1989), 515-525. doi: 10.1017/S0143385700005150.

[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.

[34]

G. Rauzy, échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.

[35]

H. SuzukiS. Ito and K. Aihara, Double rotations, Discrete Contin. Dyn. Syst., 13 (2005), 515-532. doi: 10.3934/dcds.2005.13.515.

[36]

S. Schwartzman, Asymptotic cycles, Annals of Mathematics, Second Series, 66 (1957), 270-284. doi: 10.2307/1969999.

[37]

W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. Math., 115 (1982), 201-242. doi: 10.2307/2374398.

[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.

[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.

[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.

[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.

[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.

[43]

A. Zorich, Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477-1499. doi: 10.1017/S0143385797086215.

[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.

[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.

show all references

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.

[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.

[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.

[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.

[5]

A. AvilaP. 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.

[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.

[7]

M. Bestvina and M. Feign, Stable actions of groups on real trees, Invent. Math., 121 (1995), 287-321. doi: 10.1007/BF01884300.

[8]

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

[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.

[10]

M. Boshernitzan and I. Kornfeld, Interval translation mappings, Ergodic Theory Dynam. Systems, 15 (1995), 821-832. doi: 10.1017/S0143385700009652.

[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.

[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.

[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.

[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.

[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.

[16]

C. Danthony and A. Nogueira, Measured foliations on nonorientable surfaces, Ann. Sci. école. Norm. Sup., 23 (1990), 469-494.

[17]

D. GaboriauG. 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.

[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.

[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.

[20]

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

[21]

M. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. doi: 10.1007/BF03007668.

[22]

R. Kenyon and J. Smillie, Billiards on rational-angled triangles, Comment. Math. Helv., 75 (2000), 65-108. doi: 10.1007/s000140050113.

[23]

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

[24]

G. Levitt, The dynamics of rotation pseudogroups, (French), Invent. Math., 113 (1993), 633-670. doi: 10.1007/BF01244321.

[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.

[26]

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

[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.

[28]

C. McMullen, Teichmüller geodesics of infinite complexity, Acta Math., 191 (2003), 191-223. doi: 10.1007/BF02392964.

[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.

[30]

S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, (Russian), Uspekhi Mat. Nauk, 37 (1982), 3-49.

[31]

Ch. Novak, Group actions via interval exchange transformations, PhD Thesis, Northwestern University, 2008.

[32]

A. Nogueira, Almost all interval exchange transformations with flips are nonergodic, Ergodic Theory Dynam. Systems, 9 (1989), 515-525. doi: 10.1017/S0143385700005150.

[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.

[34]

G. Rauzy, échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.

[35]

H. SuzukiS. Ito and K. Aihara, Double rotations, Discrete Contin. Dyn. Syst., 13 (2005), 515-532. doi: 10.3934/dcds.2005.13.515.

[36]

S. Schwartzman, Asymptotic cycles, Annals of Mathematics, Second Series, 66 (1957), 270-284. doi: 10.2307/1969999.

[37]

W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. Math., 115 (1982), 201-242. doi: 10.2307/2374398.

[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.

[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.

[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.

[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.

[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.

[43]

A. Zorich, Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477-1499. doi: 10.1017/S0143385797086215.

[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.

[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.

Figure 1.  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
Figure 2.  Singularities of $\mathscr F_{\pi, \mathbf a}$
Figure 3.  A transversal representing a separating cycle (bold line) and the restriction cycle (dashed line) in Example 2.18
Figure 4.  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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