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

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

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

Hieu Trung Do ^1, and Thomas A. Schmidt ^1,

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

Received March 2016 Revised September 2016 Published November 2016

Abstract
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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.

Keywords: Pisot units, Pseudo-Anosov, translation surfaces., bi-Perron units, Sah-Arnoux-Fathi invariant.

Mathematics Subject Classification: Primary: 37E30, 57M50; Secondary: 11R0.

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., Les Plans-sur-Bex, 1980), Monograph. Enseign. Math., 29, Univ. Genève, Geneva, 1981, 5-38.
[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), 97-127. 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), 199-215.
[5]	P. Arnoux and T. A. Schmidt, Veech surfaces with non-periodic directions in the trace field, J. Mod. Dyn., 3 (2009), 611-629. 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), 75-78.
[7]	J. Birman, P. Brinkmann and K. Kawamuro, Polynomial invariants of pseudo-Anosov maps, J. Topol. Anal., 4 (2012), 13-47. 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), 3433-3457. 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), 2565-2573. doi: 10.1090/proc/12958.
[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.
[11]	K. Calta and T. A. Schmidt, Infinitely many lattice surfaces with special pseudo-Anosov maps, J. Mod. Dyn., 7 (2013), 239-254. doi: 10.3934/jmd.2013.7.239.
[12]	K. Calta and J. Smillie, Algebraically periodic translation surfaces, J. Mod. Dyn., 2 (2008), 209-248. doi: 10.3934/jmd.2008.2.209.
[13]	B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012.
[14]	D. Fried, Growth rate of surface homeomorphisms and flow equivalence, Ergodic Theory and Dyn. Sys., 5 (1985), 539-563. doi: 10.1017/S0143385700003151.
[15]	J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274. doi: 10.1007/BF02395062.
[16]	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.
[17]	P. Hubert, E. Lanneau and M. Möller, The Arnoux-Yoccoz Teichmüller disc, Geom. Funct. Anal., 18 (2009), 1988-2016. 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), 631-678. 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, Contemp. Math., 532, Amer. Math. Soc., Providence, RI, 2010, 231-242. 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), 105-144. 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), 73-106. doi: 10.1080/14689360601028126.
[22]	_______, Geometric representation of interval exchange maps over algebraic number fields, Nonlinearity, 21 (2008), 149-177. doi: 10.1088/0951-7715/21/1/009.
[23]	R. Kenyon and J. Smillie, Billiards in rational-angled triangles, Comment. Mathem. Helv., 75 (2000), 65-108. doi: 10.1007/s000140050113.
[24]	D. Margalit and S. Spallone, A homological recipe for pseudo-Anosovs, Math. Res. Lett., 14 (2007), 853-863. 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), 823-872. doi: 10.1090/S0894-0347-05-00490-X.
[26]	C. T. McMullen, Teichmüller geodesics of infinite complexity, Acta Math., 191 (2003), 191-223. doi: 10.1007/BF02392964.
[27]	______, Cascades in the dynamics of measured foliations, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 1-39.
[28]	G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.
[29]	D. Rosen and C. Towse, Continued fraction representations of units associated with certain Hecke groups, Arch. Math. (Basel), 77 (2001), 294-302. doi: 10.1007/PL00000494.
[30]	B. Strenner, Lifts of pseudo-Anosov homeomorphisms of nonorientable surfaces have vanishing SAF invariant, preprint, arXiv:1604.05614.
[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.
[32]	W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.
[33]	______, Teichmüller curves in modular space, Eisenstein series and an application to triangular billiards, Inv. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.
[34]	M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100. 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, Physics, and Geometry, I, Springer, Berlin, 2006, 401-435. 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., Les Plans-sur-Bex, 1980), Monograph. Enseign. Math., 29, Univ. Genève, Geneva, 1981, 5-38.
[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), 97-127. 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), 199-215.
[5]	P. Arnoux and T. A. Schmidt, Veech surfaces with non-periodic directions in the trace field, J. Mod. Dyn., 3 (2009), 611-629. 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), 75-78.
[7]	J. Birman, P. Brinkmann and K. Kawamuro, Polynomial invariants of pseudo-Anosov maps, J. Topol. Anal., 4 (2012), 13-47. 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), 3433-3457. 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), 2565-2573. doi: 10.1090/proc/12958.
[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.
[11]	K. Calta and T. A. Schmidt, Infinitely many lattice surfaces with special pseudo-Anosov maps, J. Mod. Dyn., 7 (2013), 239-254. doi: 10.3934/jmd.2013.7.239.
[12]	K. Calta and J. Smillie, Algebraically periodic translation surfaces, J. Mod. Dyn., 2 (2008), 209-248. doi: 10.3934/jmd.2008.2.209.
[13]	B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012.
[14]	D. Fried, Growth rate of surface homeomorphisms and flow equivalence, Ergodic Theory and Dyn. Sys., 5 (1985), 539-563. doi: 10.1017/S0143385700003151.
[15]	J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274. doi: 10.1007/BF02395062.
[16]	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.
[17]	P. Hubert, E. Lanneau and M. Möller, The Arnoux-Yoccoz Teichmüller disc, Geom. Funct. Anal., 18 (2009), 1988-2016. 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), 631-678. 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, Contemp. Math., 532, Amer. Math. Soc., Providence, RI, 2010, 231-242. 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), 105-144. 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), 73-106. doi: 10.1080/14689360601028126.
[22]	_______, Geometric representation of interval exchange maps over algebraic number fields, Nonlinearity, 21 (2008), 149-177. doi: 10.1088/0951-7715/21/1/009.
[23]	R. Kenyon and J. Smillie, Billiards in rational-angled triangles, Comment. Mathem. Helv., 75 (2000), 65-108. doi: 10.1007/s000140050113.
[24]	D. Margalit and S. Spallone, A homological recipe for pseudo-Anosovs, Math. Res. Lett., 14 (2007), 853-863. 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), 823-872. doi: 10.1090/S0894-0347-05-00490-X.
[26]	C. T. McMullen, Teichmüller geodesics of infinite complexity, Acta Math., 191 (2003), 191-223. doi: 10.1007/BF02392964.
[27]	______, Cascades in the dynamics of measured foliations, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 1-39.
[28]	G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.
[29]	D. Rosen and C. Towse, Continued fraction representations of units associated with certain Hecke groups, Arch. Math. (Basel), 77 (2001), 294-302. doi: 10.1007/PL00000494.
[30]	B. Strenner, Lifts of pseudo-Anosov homeomorphisms of nonorientable surfaces have vanishing SAF invariant, preprint, arXiv:1604.05614.
[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.
[32]	W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.
[33]	______, Teichmüller curves in modular space, Eisenstein series and an application to triangular billiards, Inv. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.
[34]	M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100. 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, Physics, and Geometry, I, Springer, Berlin, 2006, 401-435. doi: 10.1007/978-3-540-31347-2\_12.

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