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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

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.  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 G. Rauzy, Représentation géométrique de suites de complexité $2n+1$, Bull. Soc. Math. France, 119 (1991), 199-215.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  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, Infinitely many lattice surfaces with special pseudo-Anosov maps, J. Mod. Dyn., 7 (2013), 239-254. doi: 10.3934/jmd.2013.7.239.  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]

B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012.  Google Scholar

[14]

D. Fried, Growth rate of surface homeomorphisms and flow equivalence, Ergodic Theory and Dyn. Sys., 5 (1985), 539-563. doi: 10.1017/S0143385700003151.  Google Scholar

[15]

J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274. doi: 10.1007/BF02395062.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

_______, 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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

______, Cascades in the dynamics of measured foliations, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 1-39.  Google Scholar

[28]

G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.  Google Scholar

[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.  Google Scholar

[30]

B. Strenner, Lifts of pseudo-Anosov homeomorphisms of nonorientable surfaces have vanishing SAF invariant,, preprint, ().   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, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.  Google Scholar

[33]

______, 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

[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.  Google Scholar

[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.  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 G. Rauzy, Représentation géométrique de suites de complexité $2n+1$, Bull. Soc. Math. France, 119 (1991), 199-215.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  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, Infinitely many lattice surfaces with special pseudo-Anosov maps, J. Mod. Dyn., 7 (2013), 239-254. doi: 10.3934/jmd.2013.7.239.  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]

B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012.  Google Scholar

[14]

D. Fried, Growth rate of surface homeomorphisms and flow equivalence, Ergodic Theory and Dyn. Sys., 5 (1985), 539-563. doi: 10.1017/S0143385700003151.  Google Scholar

[15]

J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274. doi: 10.1007/BF02395062.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

_______, 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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

______, Cascades in the dynamics of measured foliations, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 1-39.  Google Scholar

[28]

G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.  Google Scholar

[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.  Google Scholar

[30]

B. Strenner, Lifts of pseudo-Anosov homeomorphisms of nonorientable surfaces have vanishing SAF invariant,, preprint, ().   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, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.  Google Scholar

[33]

______, 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

[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.  Google Scholar

[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.  Google Scholar

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