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Weierstrass filtration on Teichmüller curves and Lyapunov exponents
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 |
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). 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. |
[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). Google Scholar |
[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). Google Scholar |
[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). 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. |
[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). 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. |
[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). Google Scholar |
[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). Google Scholar |
[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). 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. |
[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. |
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