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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 "Ergodic Theory'' (Sem., Les Plans-sur-Bex, 1980), Monograph. Enseign. Math., 29, Univ. Genève, Geneva, (1981), 5-38.  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), 97-127. 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), 611-629. 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), 75-78.  Google Scholar

[6]

W. Borho, Kettenbrüche im Galoisfeld, Abh. Math. Sem. Univ. Hamburg, 39 (1973), 76-82. 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), 83-87. 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), 139-185. 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), 1277-1298. 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), 871-908. 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), 21-42. doi: 10.1017/S1446788712000651.  Google Scholar

[12]

K. Calta and J. Smillie, Algebraically periodic translation surfaces, J. Mod. Dyn., 2 (2008), 209-248. 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). Available at http://www.cems.uvm.edu/~jvoight/research. Google Scholar

[14]

L. E. Dickson, "Linear Groups: With an Exposition of the Galois Field Theory,'' Dover Publications, Inc., New York, 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), 220-230. 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), 337-348. doi: 10.4064/aa134-4-4.  Google Scholar

[17]

P. Hubert and E. Lanneau, Veech groups without parabolic elements, Duke Math. J., 133 (2006), 335-346. 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), 461-495. 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), 73-106. doi: 10.1080/14689360601028126.  Google Scholar

[20]

_____, Geometric representation of interval-exchange maps over algebraic number fields, Nonlinearity, 21 (2008), 149-177. doi: 10.1088/0951-7715/21/1/009.  Google Scholar

[21]

W. P. Hooper, Grid graphs and lattice surfaces, preprint, arXiv:0811.0799, (2009). Google Scholar

[22]

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

[23]

A. Leutbecher, Über die Heckeschen Gruppen G($\lambda$), Abh. Math. Sem. Hamb., 31 (1967), 199-205.  Google Scholar

[24]

D. Long and A. Reid, Pseudomodular surfaces, J. Reine Angew. Math., 552 (2002), 77-100. doi: 10.1515/crll.2002.094.  Google Scholar

[25]

A. M. Macbeath, Generators of linear fractional groups, in "Number Theory'' (Proc. Symp. in Pure Math., Vol. XII, Houston, Tex., 1967) (eds. W. J. Leveque and E. G. Straus), Amer. Math. Soc., Providence, (1969), 14-32.  Google Scholar

[26]

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

[27]

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

[28]

_____, Cascades in the dynamics of measured foliations, preprint, (2012). Available from: http://www.math.harvard.edu/~ctm/papers/. Google Scholar

[29]

T. A. Schmidt and K. M. Smith, Galois orbits of principal congruence Hecke curves, J. London Math. Soc. (2), 67 (2003), 673-685. 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), 481-487. 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), 417-431. 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), 553-583. 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), 1019-1042. 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), 776-809. 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 "Ergodic Theory'' (Sem., Les Plans-sur-Bex, 1980), Monograph. Enseign. Math., 29, Univ. Genève, Geneva, (1981), 5-38.  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), 97-127. 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), 611-629. 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), 75-78.  Google Scholar

[6]

W. Borho, Kettenbrüche im Galoisfeld, Abh. Math. Sem. Univ. Hamburg, 39 (1973), 76-82. 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), 83-87. 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), 139-185. 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), 1277-1298. 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), 871-908. 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), 21-42. doi: 10.1017/S1446788712000651.  Google Scholar

[12]

K. Calta and J. Smillie, Algebraically periodic translation surfaces, J. Mod. Dyn., 2 (2008), 209-248. 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). Available at http://www.cems.uvm.edu/~jvoight/research. Google Scholar

[14]

L. E. Dickson, "Linear Groups: With an Exposition of the Galois Field Theory,'' Dover Publications, Inc., New York, 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), 220-230. 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), 337-348. doi: 10.4064/aa134-4-4.  Google Scholar

[17]

P. Hubert and E. Lanneau, Veech groups without parabolic elements, Duke Math. J., 133 (2006), 335-346. 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), 461-495. 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), 73-106. doi: 10.1080/14689360601028126.  Google Scholar

[20]

_____, Geometric representation of interval-exchange maps over algebraic number fields, Nonlinearity, 21 (2008), 149-177. doi: 10.1088/0951-7715/21/1/009.  Google Scholar

[21]

W. P. Hooper, Grid graphs and lattice surfaces, preprint, arXiv:0811.0799, (2009). Google Scholar

[22]

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

[23]

A. Leutbecher, Über die Heckeschen Gruppen G($\lambda$), Abh. Math. Sem. Hamb., 31 (1967), 199-205.  Google Scholar

[24]

D. Long and A. Reid, Pseudomodular surfaces, J. Reine Angew. Math., 552 (2002), 77-100. doi: 10.1515/crll.2002.094.  Google Scholar

[25]

A. M. Macbeath, Generators of linear fractional groups, in "Number Theory'' (Proc. Symp. in Pure Math., Vol. XII, Houston, Tex., 1967) (eds. W. J. Leveque and E. G. Straus), Amer. Math. Soc., Providence, (1969), 14-32.  Google Scholar

[26]

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

[27]

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

[28]

_____, Cascades in the dynamics of measured foliations, preprint, (2012). Available from: http://www.math.harvard.edu/~ctm/papers/. Google Scholar

[29]

T. A. Schmidt and K. M. Smith, Galois orbits of principal congruence Hecke curves, J. London Math. Soc. (2), 67 (2003), 673-685. 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), 481-487. 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), 417-431. 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), 553-583. 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), 1019-1042. 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), 776-809. doi: 10.1007/s00039-013-0221-z.  Google Scholar

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