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April  2013, 7(2): 239-254. doi: 10.3934/jmd.2013.7.239

Infinitely many lattice surfaces with special pseudo-Anosov maps

Kariane Calta 1, and  Thomas A. Schmidt 2,

1. 

Vassar College, Poughkeepsie, NY 12604-0257, United States
2. 

Oregon State University, Corvallis, OR 97331, United States

Received  October 2012 Revised  May 2013 Published  September 2013

We give explicit pseudo-Anosov homeomorphisms with vanishing Sah-Arnoux-Fathi invariant. Any translation surface whose Veech group is commensurable to any of a large class of triangle groups is shown to have an affine pseudo-Anosov homeomorphism of this type. We also apply a reduction to finite triangle groups and thereby show the existence of nonparabolic elements in the periodic field of certain translation surfaces.
Keywords: translation surface, flux, Veech group., Pseudo-Anosov, SAF invariant.
Mathematics Subject Classification: Primary: 37D99; Secondary: 30F60, 37A25, 37A45, 37F99, 58F1.
Citation: Kariane Calta, Thomas A. Schmidt. Infinitely many lattice surfaces with special pseudo-Anosov maps. Journal of Modern Dynamics, 2013, 7 (2) : 239-254. doi: 10.3934/jmd.2013.7.239
References:
[1]

P. Arnoux, Échanges d'intervalles et flots sur les surfaces,, in, 29 (1981), 5. Google Scholar

[2]

_____, "Thèse de 3$^e$ Cycle,'', Université de Reims, (1981). Google Scholar

[3]

P. Arnoux, J. Bernat and X. Bressaud, Geometrical models for substitutions,, Exp. Math., 20 (2011), 97. doi: 10.1080/10586458.2011.544590. Google Scholar

[4]

P. Arnoux and T. A. Schmidt, Veech surfaces with nonperiodic directions in the trace field,, J. Mod. Dyn., 3 (2009), 611. doi: 10.3934/jmd.2009.3.611. Google Scholar

[5]

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

[6]

W. Borho, Kettenbrüche im Galoisfeld,, Abh. Math. Sem. Univ. Hamburg, 39 (1973), 76. doi: 10.1007/BF02992820. Google Scholar

[7]

W. Borho and G. Rosenberger, Eine Bemerkung zur Hecke-Gruppe $G(\lambda )$,, Abh. Math. Sem. Univ. Hamburg, 39 (1973), 83. doi: 10.1007/BF02992821. Google Scholar

[8]

I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents,, Ann. of Math. (2), 172 (2010), 139. doi: 10.4007/annals.2010.172.139. Google Scholar

[9]

R. M. Burton, C. Kraaikamp and T. A. Schmidt, Natural extensions for the Rosen fractions,, Trans. Amer. Math. Soc., 352 (1999), 1277. doi: 10.1090/S0002-9947-99-02442-3. Google Scholar

[10]

K. Calta, Veech surfaces and complete periodicity in genus two,, J. Amer. Math. Soc., 17 (2004), 871. doi: 10.1090/S0894-0347-04-00461-8. Google Scholar

[11]

K. Calta and T. A. Schmidt, Continued fractions for a class of triangle groups,, J. Austral. Math. Soc., 93 (2012), 21. doi: 10.1017/S1446788712000651. Google Scholar

[12]

K. Calta and J. Smillie, Algebraically periodic translation surfaces,, J. Mod. Dyn., 2 (2008), 209. doi: 10.3934/jmd.2008.2.209. Google Scholar

[13]

P. L. Clark and J. Voight, Algebraic curves uniformized by congruence subgroups of triangle groups,, preprint, (2011). Google Scholar

[14]

L. E. Dickson, "Linear Groups: With an Exposition of the Galois Field Theory,'', Dover Publications, (1958). Google Scholar

[15]

M.-L. Lang, C.-H. Lim and S.-P. Tan, Principal congruence subgroups of the Hecke groups,, J. Number Th., 85 (2000), 220. doi: 10.1006/jnth.2000.2542. Google Scholar

[16]

E. Hanson, A. Merberg, C. Towse and E. Yudovina, Generalized continued fractions and orbits under the action of Hecke triangle groups,, Acta Arith., 134 (2008), 337. doi: 10.4064/aa134-4-4. Google Scholar

[17]

P. Hubert and E. Lanneau, Veech groups without parabolic elements,, Duke Math. J., 133 (2006), 335. doi: 10.1215/S0012-7094-06-13326-4. Google Scholar

[18]

P. Hubert and T. A. Schmidt, Invariants of translation surfaces,, Ann. Inst. Fourier (Grenoble), 51 (2001), 461. doi: 10.5802/aif.1829. Google Scholar

[19]

J. H. Lowenstein, G. Poggiaspalla and F. Vivaldi, Interval-exchange transformations over algebraic number fields: The cubic Arnoux-Yoccoz model,, Dyn. Syst., 22 (2007), 73. doi: 10.1080/14689360601028126. Google Scholar

[20]

_____, Geometric representation of interval-exchange maps over algebraic number fields,, Nonlinearity, 21 (2008), 149. doi: 10.1088/0951-7715/21/1/009. Google Scholar

[21]

W. P. Hooper, Grid graphs and lattice surfaces,, preprint, (2009). Google Scholar

[22]

R. Kenyon and J. Smillie, Billiards in rational-angled triangles,, Comment. Mathem. Helv., 75 (2000), 65. doi: 10.1007/s000140050113. Google Scholar

[23]

A. Leutbecher, Über die Heckeschen Gruppen G($\lambda$),, Abh. Math. Sem. Hamb., 31 (1967), 199. Google Scholar

[24]

D. Long and A. Reid, Pseudomodular surfaces,, J. Reine Angew. Math., 552 (2002), 77. doi: 10.1515/crll.2002.094. Google Scholar

[25]

A. M. Macbeath, Generators of linear fractional groups,, in, (1969), 14. Google Scholar

[26]

C. Maclachlan and A. Reid, "The Arithmetic of Hyperbolic 3-Manifolds,'', Graduate Texts in Mathematics, 219 (2003). Google Scholar

[27]

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

[28]

_____, Cascades in the dynamics of measured foliations,, preprint, (2012). Google Scholar

[29]

T. A. Schmidt and K. M. Smith, Galois orbits of principal congruence Hecke curves,, J. London Math. Soc. (2), 67 (2003), 673. doi: 10.1112/S0024610703004113. Google Scholar

[30]

L. A. Parson, Normal congruence subgroups of the Hecke groups $G(2^{1/2})$ and $G(3^{1/2})$,, Pacific J. Math., 70 (1977), 481. doi: 10.2140/pjm.1977.70.481. Google Scholar

[31]

W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces,, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417. doi: 10.1090/S0273-0979-1988-15685-6. Google Scholar

[32]

W. A. Veech, Teichmüller curves in modular space, Eisenstein series, and an application to triangular billiards,, Inv. Math., 97 (1989), 553. doi: 10.1007/BF01388890. Google Scholar

[33]

C. Ward, Calculation of Fuchsian groups associated to billiards in a rational triangle,, Ergodic Theory Dynam. Systems, 18 (1998), 1019. doi: 10.1017/S0143385798117479. Google Scholar

[34]

A. Wright, Schwartz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller curves,, Geom. Funct. Anal., 23 (2013), 776. doi: 10.1007/s00039-013-0221-z. Google Scholar

show all references

References:
[1]

P. Arnoux, Échanges d'intervalles et flots sur les surfaces,, in, 29 (1981), 5. Google Scholar

[2]

_____, "Thèse de 3$^e$ Cycle,'', Université de Reims, (1981). Google Scholar

[3]

P. Arnoux, J. Bernat and X. Bressaud, Geometrical models for substitutions,, Exp. Math., 20 (2011), 97. doi: 10.1080/10586458.2011.544590. Google Scholar

[4]

P. Arnoux and T. A. Schmidt, Veech surfaces with nonperiodic directions in the trace field,, J. Mod. Dyn., 3 (2009), 611. doi: 10.3934/jmd.2009.3.611. Google Scholar

[5]

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

[6]

W. Borho, Kettenbrüche im Galoisfeld,, Abh. Math. Sem. Univ. Hamburg, 39 (1973), 76. doi: 10.1007/BF02992820. Google Scholar

[7]

W. Borho and G. Rosenberger, Eine Bemerkung zur Hecke-Gruppe $G(\lambda )$,, Abh. Math. Sem. Univ. Hamburg, 39 (1973), 83. doi: 10.1007/BF02992821. Google Scholar

[8]

I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents,, Ann. of Math. (2), 172 (2010), 139. doi: 10.4007/annals.2010.172.139. Google Scholar

[9]

R. M. Burton, C. Kraaikamp and T. A. Schmidt, Natural extensions for the Rosen fractions,, Trans. Amer. Math. Soc., 352 (1999), 1277. doi: 10.1090/S0002-9947-99-02442-3. Google Scholar

[10]

K. Calta, Veech surfaces and complete periodicity in genus two,, J. Amer. Math. Soc., 17 (2004), 871. doi: 10.1090/S0894-0347-04-00461-8. Google Scholar

[11]

K. Calta and T. A. Schmidt, Continued fractions for a class of triangle groups,, J. Austral. Math. Soc., 93 (2012), 21. doi: 10.1017/S1446788712000651. Google Scholar

[12]

K. Calta and J. Smillie, Algebraically periodic translation surfaces,, J. Mod. Dyn., 2 (2008), 209. doi: 10.3934/jmd.2008.2.209. Google Scholar

[13]

P. L. Clark and J. Voight, Algebraic curves uniformized by congruence subgroups of triangle groups,, preprint, (2011). Google Scholar

[14]

L. E. Dickson, "Linear Groups: With an Exposition of the Galois Field Theory,'', Dover Publications, (1958). Google Scholar

[15]

M.-L. Lang, C.-H. Lim and S.-P. Tan, Principal congruence subgroups of the Hecke groups,, J. Number Th., 85 (2000), 220. doi: 10.1006/jnth.2000.2542. Google Scholar

[16]

E. Hanson, A. Merberg, C. Towse and E. Yudovina, Generalized continued fractions and orbits under the action of Hecke triangle groups,, Acta Arith., 134 (2008), 337. doi: 10.4064/aa134-4-4. Google Scholar

[17]

P. Hubert and E. Lanneau, Veech groups without parabolic elements,, Duke Math. J., 133 (2006), 335. doi: 10.1215/S0012-7094-06-13326-4. Google Scholar

[18]

P. Hubert and T. A. Schmidt, Invariants of translation surfaces,, Ann. Inst. Fourier (Grenoble), 51 (2001), 461. doi: 10.5802/aif.1829. Google Scholar

[19]

J. H. Lowenstein, G. Poggiaspalla and F. Vivaldi, Interval-exchange transformations over algebraic number fields: The cubic Arnoux-Yoccoz model,, Dyn. Syst., 22 (2007), 73. doi: 10.1080/14689360601028126. Google Scholar

[20]

_____, Geometric representation of interval-exchange maps over algebraic number fields,, Nonlinearity, 21 (2008), 149. doi: 10.1088/0951-7715/21/1/009. Google Scholar

[21]

W. P. Hooper, Grid graphs and lattice surfaces,, preprint, (2009). Google Scholar

[22]

R. Kenyon and J. Smillie, Billiards in rational-angled triangles,, Comment. Mathem. Helv., 75 (2000), 65. doi: 10.1007/s000140050113. Google Scholar

[23]

A. Leutbecher, Über die Heckeschen Gruppen G($\lambda$),, Abh. Math. Sem. Hamb., 31 (1967), 199. Google Scholar

[24]

D. Long and A. Reid, Pseudomodular surfaces,, J. Reine Angew. Math., 552 (2002), 77. doi: 10.1515/crll.2002.094. Google Scholar

[25]

A. M. Macbeath, Generators of linear fractional groups,, in, (1969), 14. Google Scholar

[26]

C. Maclachlan and A. Reid, "The Arithmetic of Hyperbolic 3-Manifolds,'', Graduate Texts in Mathematics, 219 (2003). Google Scholar

[27]

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

[28]

_____, Cascades in the dynamics of measured foliations,, preprint, (2012). Google Scholar

[29]

T. A. Schmidt and K. M. Smith, Galois orbits of principal congruence Hecke curves,, J. London Math. Soc. (2), 67 (2003), 673. doi: 10.1112/S0024610703004113. Google Scholar

[30]

L. A. Parson, Normal congruence subgroups of the Hecke groups $G(2^{1/2})$ and $G(3^{1/2})$,, Pacific J. Math., 70 (1977), 481. doi: 10.2140/pjm.1977.70.481. Google Scholar

[31]

W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces,, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417. doi: 10.1090/S0273-0979-1988-15685-6. Google Scholar

[32]

W. A. Veech, Teichmüller curves in modular space, Eisenstein series, and an application to triangular billiards,, Inv. Math., 97 (1989), 553. doi: 10.1007/BF01388890. Google Scholar

[33]

C. Ward, Calculation of Fuchsian groups associated to billiards in a rational triangle,, Ergodic Theory Dynam. Systems, 18 (1998), 1019. doi: 10.1017/S0143385798117479. Google Scholar

[34]

A. Wright, Schwartz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller curves,, Geom. Funct. Anal., 23 (2013), 776. doi: 10.1007/s00039-013-0221-z. Google Scholar

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