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

Infinitely many lattice surfaces with special pseudo-Anosov maps

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.
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.

[2]

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

[3]

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

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[2]

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

[3]

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

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[1]

Hieu Trung Do, Thomas A. Schmidt. New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant. Journal of Modern Dynamics, 2016, 10: 541-561. doi: 10.3934/jmd.2016.10.541

[2]

Chris Johnson, Martin Schmoll. Pseudo-Anosov eigenfoliations on Panov planes. Electronic Research Announcements, 2014, 21: 89-108. doi: 10.3934/era.2014.21.89

[3]

S. Öykü Yurttaş. Dynnikov and train track transition matrices of pseudo-Anosov braids. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 541-570. doi: 10.3934/dcds.2016.36.541

[4]

Pascal Hubert, Gabriela Schmithüsen. Infinite translation surfaces with infinitely generated Veech groups. Journal of Modern Dynamics, 2010, 4 (4) : 715-732. doi: 10.3934/jmd.2010.4.715

[5]

Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699

[6]

Eric Bedford, Serge Cantat, Kyounghee Kim. Pseudo-automorphisms with no invariant foliation. Journal of Modern Dynamics, 2014, 8 (2) : 221-250. doi: 10.3934/jmd.2014.8.221

[7]

Juan Alonso, Nancy Guelman, Juliana Xavier. Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo)-Anosov elements. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1817-1827. doi: 10.3934/dcds.2015.35.1817

[8]

Patrick Foulon, Boris Hasselblatt. Lipschitz continuous invariant forms for algebraic Anosov systems. Journal of Modern Dynamics, 2010, 4 (3) : 571-584. doi: 10.3934/jmd.2010.4.571

[9]

David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477

[10]

Rafael De La Llave, Victoria Sadovskaya. On the regularity of integrable conformal structures invariant under Anosov systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 377-385. doi: 10.3934/dcds.2005.12.377

[11]

Lennard F. Bakker, Pedro Martins Rodrigues. A profinite group invariant for hyperbolic toral automorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1965-1976. doi: 10.3934/dcds.2012.32.1965

[12]

S. A. Krat. On pairs of metrics invariant under a cocompact action of a group. Electronic Research Announcements, 2001, 7: 79-86.

[13]

Jamshid Moori, Amin Saeidi. Some designs and codes invariant under the Tits group. Advances in Mathematics of Communications, 2017, 11 (1) : 77-82. doi: 10.3934/amc.2017003

[14]

John Franks, Michael Handel. Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of a surface. Journal of Modern Dynamics, 2013, 7 (3) : 369-394. doi: 10.3934/jmd.2013.7.369

[15]

Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. Journal of Geometric Mechanics, 2014, 6 (3) : 335-372. doi: 10.3934/jgm.2014.6.335

[16]

Hans Koch. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 881-909. doi: 10.3934/dcds.2004.11.881

[17]

Noreen Sher Akbar, Dharmendra Tripathi, Zafar Hayat Khan. Numerical investigation of Cattanneo-Christov heat flux in CNT suspended nanofluid flow over a stretching porous surface with suction and injection. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 583-594. doi: 10.3934/dcdss.2018033

[18]

Bernardo Gabriel Rodrigues. Some optimal codes related to graphs invariant under the alternating group $A_8$. Advances in Mathematics of Communications, 2011, 5 (2) : 339-350. doi: 10.3934/amc.2011.5.339

[19]

Yoshikazu Katayama, Colin E. Sutherland and Masamichi Takesaki. The intrinsic invariant of an approximately finite dimensional factor and the cocycle conjugacy of discrete amenable group actions. Electronic Research Announcements, 1995, 1: 43-47.

[20]

Xiankun Ren. Periodic measures are dense in invariant measures for residually finite amenable group actions with specification. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1657-1667. doi: 10.3934/dcds.2018068

2016 Impact Factor: 0.706

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]