2016, 10: 541-561. doi: 10.3934/jmd.2016.10.541

New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant

1. 

Department of Mathematics, Oregon State University, Corvallis, OR 97331, United States

Received  March 2016 Revised  September 2016 Published  November 2016

We show that an orientable pseudo-Anosov homeomorphism has vanishing Sah-Arnoux-Fathi invariant if and only if the minimal polynomial of its dilatation is not reciprocal. We relate this to works of Margalit-Spallone and Birman, Brinkmann and Kawamuro. Mainly, we use Veech's construction of pseudo-Anosov maps to give explicit pseudo-Anosov maps of vanishing Sah-Arnoux-Fathi invariant. In particular, we give a new infinite family of such maps in genus 3.
Citation: 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
References:
[1]

P. Arnoux, Échanges d'intervalles et flots sur les surfaces,, in Ergodic Theory (Sem., (1980), 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 G. Rauzy, Représentation géométrique de suites de complexité $2n+1$,, Bull. Soc. Math. France, 119 (1991), 199.

[5]

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

[6]

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

[7]

J. Birman, P. Brinkmann and K. Kawamuro, Polynomial invariants of pseudo-Anosov maps,, J. Topol. Anal., 4 (2012), 13. doi: 10.1142/S1793525312500033.

[8]

C. Boissy, Classification of Rauzy classes in the moduli space of abelian and quadratic differentials,, Discrete Contin. Dyn. Syst., 32 (2012), 3433. doi: 10.3934/dcds.2012.32.3433.

[9]

M. Boshernitzan, Subgroup of interval exchanges generated by torsion elements and rotations,, Proc. Amer. Math. Soc., 144 (2016), 2565. doi: 10.1090/proc/12958.

[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, Infinitely many lattice surfaces with special pseudo-Anosov maps,, J. Mod. Dyn., 7 (2013), 239. doi: 10.3934/jmd.2013.7.239.

[12]

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

[13]

B. Farb and D. Margalit, A Primer on Mapping Class Groups,, Princeton Mathematical Series, (2012).

[14]

D. Fried, Growth rate of surface homeomorphisms and flow equivalence,, Ergodic Theory and Dyn. Sys., 5 (1985), 539. doi: 10.1017/S0143385700003151.

[15]

J. Hubbard and H. Masur, Quadratic differentials and foliations,, Acta Math., 142 (1979), 221. doi: 10.1007/BF02395062.

[16]

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

[17]

P. Hubert, E. Lanneau and M. Möller, The Arnoux-Yoccoz Teichmüller disc,, Geom. Funct. Anal., 18 (2009), 1988. doi: 10.1007/s00039-009-0706-y.

[18]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,, Invent. Math., 153 (2003), 631. doi: 10.1007/s00222-003-0303-x.

[19]

E. Lanneau, Infinite sequence of fixed point free pseudo-Anosov homeomorphisms on a family of genus two surfaces,, in Dynamical Numbers - Interplay Between Dynamical Systems and Number Theory, (2010), 231. doi: 10.1090/conm/532/10493.

[20]

E. Lanneau and J.-C. Thiffeault, On the minimum dilatation of pseudo-Anosov homeomorphisms on surfaces of small genus,, Ann. Inst. Fourier (Grenoble), 61 (2011), 105. doi: 10.5802/aif.2599.

[21]

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.

[22]

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

[23]

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

[24]

D. Margalit and S. Spallone, A homological recipe for pseudo-Anosovs,, Math. Res. Lett., 14 (2007), 853. doi: 10.4310/MRL.2007.v14.n5.a12.

[25]

S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps,, J. Amer. Math. Soc., 18 (2005), 823. doi: 10.1090/S0894-0347-05-00490-X.

[26]

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

[27]

______, Cascades in the dynamics of measured foliations,, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 1.

[28]

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

[29]

D. Rosen and C. Towse, Continued fraction representations of units associated with certain Hecke groups,, Arch. Math. (Basel), 77 (2001), 294. doi: 10.1007/PL00000494.

[30]

B. Strenner, Lifts of pseudo-Anosov homeomorphisms of nonorientable surfaces have vanishing SAF invariant,, preprint, ().

[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, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201. doi: 10.2307/1971391.

[33]

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

[34]

M. Viana, Ergodic theory of interval exchange maps,, Rev. Mat. Complut., 19 (2006), 7. doi: 10.5209/rev_REMA.2006.v19.n1.16621.

[35]

J.-C. Yoccoz, Continued fraction algorithms for interval exchange maps: An introduction,, in Frontiers in Number Theory, (2006), 401. doi: 10.1007/978-3-540-31347-2\_12.

show all references

References:
[1]

P. Arnoux, Échanges d'intervalles et flots sur les surfaces,, in Ergodic Theory (Sem., (1980), 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 G. Rauzy, Représentation géométrique de suites de complexité $2n+1$,, Bull. Soc. Math. France, 119 (1991), 199.

[5]

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

[6]

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

[7]

J. Birman, P. Brinkmann and K. Kawamuro, Polynomial invariants of pseudo-Anosov maps,, J. Topol. Anal., 4 (2012), 13. doi: 10.1142/S1793525312500033.

[8]

C. Boissy, Classification of Rauzy classes in the moduli space of abelian and quadratic differentials,, Discrete Contin. Dyn. Syst., 32 (2012), 3433. doi: 10.3934/dcds.2012.32.3433.

[9]

M. Boshernitzan, Subgroup of interval exchanges generated by torsion elements and rotations,, Proc. Amer. Math. Soc., 144 (2016), 2565. doi: 10.1090/proc/12958.

[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, Infinitely many lattice surfaces with special pseudo-Anosov maps,, J. Mod. Dyn., 7 (2013), 239. doi: 10.3934/jmd.2013.7.239.

[12]

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

[13]

B. Farb and D. Margalit, A Primer on Mapping Class Groups,, Princeton Mathematical Series, (2012).

[14]

D. Fried, Growth rate of surface homeomorphisms and flow equivalence,, Ergodic Theory and Dyn. Sys., 5 (1985), 539. doi: 10.1017/S0143385700003151.

[15]

J. Hubbard and H. Masur, Quadratic differentials and foliations,, Acta Math., 142 (1979), 221. doi: 10.1007/BF02395062.

[16]

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

[17]

P. Hubert, E. Lanneau and M. Möller, The Arnoux-Yoccoz Teichmüller disc,, Geom. Funct. Anal., 18 (2009), 1988. doi: 10.1007/s00039-009-0706-y.

[18]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,, Invent. Math., 153 (2003), 631. doi: 10.1007/s00222-003-0303-x.

[19]

E. Lanneau, Infinite sequence of fixed point free pseudo-Anosov homeomorphisms on a family of genus two surfaces,, in Dynamical Numbers - Interplay Between Dynamical Systems and Number Theory, (2010), 231. doi: 10.1090/conm/532/10493.

[20]

E. Lanneau and J.-C. Thiffeault, On the minimum dilatation of pseudo-Anosov homeomorphisms on surfaces of small genus,, Ann. Inst. Fourier (Grenoble), 61 (2011), 105. doi: 10.5802/aif.2599.

[21]

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.

[22]

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

[23]

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

[24]

D. Margalit and S. Spallone, A homological recipe for pseudo-Anosovs,, Math. Res. Lett., 14 (2007), 853. doi: 10.4310/MRL.2007.v14.n5.a12.

[25]

S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps,, J. Amer. Math. Soc., 18 (2005), 823. doi: 10.1090/S0894-0347-05-00490-X.

[26]

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

[27]

______, Cascades in the dynamics of measured foliations,, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 1.

[28]

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

[29]

D. Rosen and C. Towse, Continued fraction representations of units associated with certain Hecke groups,, Arch. Math. (Basel), 77 (2001), 294. doi: 10.1007/PL00000494.

[30]

B. Strenner, Lifts of pseudo-Anosov homeomorphisms of nonorientable surfaces have vanishing SAF invariant,, preprint, ().

[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, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201. doi: 10.2307/1971391.

[33]

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

[34]

M. Viana, Ergodic theory of interval exchange maps,, Rev. Mat. Complut., 19 (2006), 7. doi: 10.5209/rev_REMA.2006.v19.n1.16621.

[35]

J.-C. Yoccoz, Continued fraction algorithms for interval exchange maps: An introduction,, in Frontiers in Number Theory, (2006), 401. doi: 10.1007/978-3-540-31347-2\_12.

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