Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices

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February 2013, 33(2): 527-553. doi: 10.3934/dcds.2013.33.527

Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices

Michael Baake ^1, , Natascha Neumärker ^1, and John A. G. Roberts ^2,

1.	Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany, Germany
2.	School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

Received August 2011 Revised May 2012 Published September 2012

Abstract
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Under a Creative Commons license

We study various aspects of the dynamics induced by integer matrices on the invariant rational lattices of the torus in dimension $2$ and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on the lattice, characterising the pretails of eventually periodic orbits. Next we study the nature of the symmetries and reversing symmetries of toral automorphisms on a given lattice, which has particular relevance to (quantum) cat maps.

Keywords: symmetry, toral automorphism, reversing symmetry., periodic orbit, Toral endomorphism, orbit structure.

Mathematics Subject Classification: 37E30, 37E1.

Citation: Michael Baake, Natascha Neumärker, John A. G. Roberts. Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 527-553. doi: 10.3934/dcds.2013.33.527

References:

[1]	W. A. Adkins and S. H. Weintraub, "Algebra - An Approach via Module Theory," corr. 2nd printing, Springer, New York, 1999.
[2]	R. Adler, C. Tresser and P. A. Worfolk, Topological conjugacy of linear endomorphisms of the 2-torus, Trans. AMS, 349 (1997), 1633-1652. doi: 10.1090/S0002-9947-97-01895-3.
[3]	V. I. Arnold and A. Avez, "Ergodic Problems of Classical Mechanics," reprint, Addison-Wesley, Redwood City, CA, 1989.
[4]	N. Avni, U. Onn, A. Prasad and L. Vaserstein, Similarity classes of $3\times 3$ matrices over a localprincipal ideal ring, Commun. Algebra, 37 (2009), 2601-2615. doi: 10.1080/00927870902747266.
[5]	H. Aydin, R. Dikici and G. C. Smith, Wall and Vinston revisited, in "Applications of Fibonacci Numbers," 5 (1992), K 61-68. Kluwer Acad. Publ., Dordrecht, 1993.
[6]	M. Baake, U. Grimm and D. Joseph, Trace maps, invariants, and some of their applications, Int. J. Mod. Phys. B, 7 (1993), 1527-1550. doi: 10.1142/S021797929300247X.
[7]	M. Baake, J. Hermisson and P. A. B. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A: Math. Gen., 30 (1997), 3029-3056.
[8]	M. Baake, E. Lau and V. Paskunas, A note on the dynamical zeta function of general toral endomorphisms, Monatsh. Math., 161 (2010), 33-42.
[9]	M. Baake and J. A. G. Roberts, Reversing symmetry group of $\GL(2,\ZZ)$ and $\PGL(2,\ZZ)$matrices with connections to cat maps and trace maps, J. Phys. A: Math. Gen., 30 (1997), 1549-1573. doi: 10.1088/0305-4470/30/5/020.
[10]	M. Baake and J. A . G. Roberts, symmetries of toral automorphisms, Nonlinearity, 14 (2001), R1-R24.
[11]	M. Baake and J. A. G. Roberts, The structure of reversing symmetry groups, Bull. Austral. Math. Soc., 73 (2006), 445-459.
[12]	M. Baake, J. A. G. Roberts and A. Weiss, Periodic orbits of linear endomorphisms of the $2$-torusand its lattices, Nonlinearity, 21 (2008), 2427-2446. doi: 10.1088/0951-7715/21/10/012.
[13]	E. Behrends and B. Fiedler, Periods of discretized linear Anosov maps, Ergod. Th. & Dynam. Syst., 18 (1998), 331-341. doi: 10.1017/S0143385798100366.
[14]	E. Brown and T. P. Vaughan, Cycles of directed graphs definedby matrix multiplication (mod $n$), Discr. Math., 239 (2001), 109-120. doi: 10.1016/S0012-365X(01)00040-1.
[15]	P. Bundschuh and J. S. Shiue, A generalization of a paper by D. D. Wall, Rendiconti Accademia Nazionale dei Lincei, Roma,Classe di Scienze Fisiche,Matematiche e Naturali, 56 (1974), 135-144.
[16]	D. Damanik, Gordon-type arguments in the spectral theory of one-dimensional quasicrystals, in "Directions in Mathematical Quasicrystals" (eds. M. Baake and R. V. Moody), CRM Monograph Series, AMS, Providence, RI, 13 (2000), 277-305.
[17]	R. W. Davis, Certain matrix equations over rings of integers, Duke Math. J., 35 (1968), 49-59. doi: 10.1215/S0012-7094-68-03506-0.
[18]	M. Degli Esposti and S. Isola, Distribution of closed orbits for linear automorphisms of tori, Nonlinearity, 8 (1995), 827-842. doi: 10.1088/0951-7715/8/5/010.
[19]	M. Degli Esposti and B. Winn, The quantum perturbed cat map and symmetry, J. Phys. A: Math. Gen., 38 (2005), 5895-5912. doi: 10.1088/0305-4470/38/26/005.
[20]	R. De Vogelaere, On the structure of symmetric periodic solutions of conservative systems, with applications, Ch. IV of Contributions to the Theory of NonlinearOscillations, ed. S. Lefschetz, Princeton Univ. Press, Princeton, IV 1958, 53-84.
[21]	F. J. Dyson and H. Falk, Period of a discrete cat mapping, Amer. Math. Monthly, 99 (1992), 603-614. doi: 10.2307/2324989.
[22]	H. T. Engstrom, On sequences defined by linear recurrence relations, Trans. Amer. Math. Soc., 33 (1931), 210-218. doi: 10.1090/S0002-9947-1931-1501585-5.
[23]	A. Fel'shtyn, Dynamical zeta functions, nielsen theory and reidemeister torsion, Memoirs AMS, Providence, RI, 147 (2000), xii+146 pp.
[24]	F. R. Gantmacher, "Matrix Theory," Chelsea, New York, I 1960.
[25]	G. Gaspari, The Arnold cat map on prime lattices, Physica, 73 (1994), 352-372.
[26]	H. Hasse, "Number Theory," Springer, Berlin 1980. doi: 10.1007/978-3-642-66671-1.
[27]	N. Jacobson, "Lectures in Abstract Algebra. II. Linear Algebra," reprint, Springer, New York, 1975.
[28]	A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995.
[29]	J. P. Keating, Asymptotic properties of the periodic orbits of the cat maps, Nonlinearity, 4 (1991), 277-307. doi: 10.1088/0951-7715/4/2/005.
[30]	J. P. Keating and F. Mezzadri, Pseudo-symmetries of Anosov maps and spectral statistics, Nonlinearity, 13 (2000), 747-775. doi: 10.1088/0951-7715/13/3/313.
[31]	P. Kurlberg, On the order of unimodular matrices modulo integers, Acta Arithm., 110 (2003), 141-151. doi: 10.4064/aa110-2-4.
[32]	P. Kurlberg and Z. Rudnick, Hecke theory and equidistribution for the quantization of linear maps of the torus, Duke Math. J., 103 (2000), 47-77. doi: 10.1215/S0012-7094-00-10314-6.
[33]	S. Lang, "Algebra," rev. 3rd ed., Springer, New York, 2002. doi: 10.1007/978-1-4613-0041-0.
[34]	R. Lidl and H. Niederreiter, "Introduction to Finite Fields and Their Applications," Cambridge University Press, Cambridge, 1986.
[35]	N. Neumärker, "Orbitstatistik und Relative Realisierbarkeit," Diploma Thesis, Univ. Bielefeld, 2007.
[36]	N. Neumärker, "The Arithmetic Structure of Discrete Dynamical Systems on the Torus," PhD thesis, Univ. Bielefeld, 2012.
[37]	I. Percival and F. Vivaldi, Arithmetical properties of strongly chaotic motions, Physica, 25 (1987), 105-130.
[38]	J. A. G. Roberts and M. Baake, Trace maps as 3D reversible dynamical systems with an invariant, J. Stat. Phys., 74 (1994), 829-888. doi: 10.1007/BF02188581.
[39]	J. A. G. Roberts and G. R. W. Quispel, Chaos and time-reversal symmetry. Order and chaos inreversible dynamical systems, Phys. Rep., 216 (1992), 63-177. doi: 10.1016/0370-1573(92)90163-T.
[40]	D. Ruelle, "Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval," CRM Monograph Series, AMS, Providence, RI, 4 1994.
[41]	P. Seibt, A period formula for torus automorphisms, Discr. Cont. Dynam. Syst., 9 (2003), 1029-1048. doi: 10.3934/dcds.2003.9.1029.
[42]	N. J. A. Sloane, "The Online Encyclopedia of Integer Sequences," http://oeis.org
[43]	O. Taussky, Introduction into connections between algebraic number theory and integral matrices, 2nd appendix to: H. Cohn, A Classical Invitation to Algebraic Numbers and Class Fields, 2nd printing, Springer,New York, (1988), 305-321.
[44]	D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532. doi: 10.2307/2309169.
[45]	P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.
[46]	M. Ward, The arithmetic theory of linear recurring sequences, Trans. Amer. Math. Soc., 35 (1933), 600-628. doi: 10.1090/S0002-9947-1933-1501705-4.
[47]	R. J. Wilson, "Introduction to Graph Theory," 4th ed., Prentice Hall, Harlow, 1996.
[48]	N. Zierler, Linear recurring sequences, J. Soc. Indust. Appl. Math., 7 (1959), 31-48. doi: 10.1137/0107003.

show all references

References:

[1]	W. A. Adkins and S. H. Weintraub, "Algebra - An Approach via Module Theory," corr. 2nd printing, Springer, New York, 1999.
[2]	R. Adler, C. Tresser and P. A. Worfolk, Topological conjugacy of linear endomorphisms of the 2-torus, Trans. AMS, 349 (1997), 1633-1652. doi: 10.1090/S0002-9947-97-01895-3.
[3]	V. I. Arnold and A. Avez, "Ergodic Problems of Classical Mechanics," reprint, Addison-Wesley, Redwood City, CA, 1989.
[4]	N. Avni, U. Onn, A. Prasad and L. Vaserstein, Similarity classes of $3\times 3$ matrices over a localprincipal ideal ring, Commun. Algebra, 37 (2009), 2601-2615. doi: 10.1080/00927870902747266.
[5]	H. Aydin, R. Dikici and G. C. Smith, Wall and Vinston revisited, in "Applications of Fibonacci Numbers," 5 (1992), K 61-68. Kluwer Acad. Publ., Dordrecht, 1993.
[6]	M. Baake, U. Grimm and D. Joseph, Trace maps, invariants, and some of their applications, Int. J. Mod. Phys. B, 7 (1993), 1527-1550. doi: 10.1142/S021797929300247X.
[7]	M. Baake, J. Hermisson and P. A. B. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A: Math. Gen., 30 (1997), 3029-3056.
[8]	M. Baake, E. Lau and V. Paskunas, A note on the dynamical zeta function of general toral endomorphisms, Monatsh. Math., 161 (2010), 33-42.
[9]	M. Baake and J. A. G. Roberts, Reversing symmetry group of $\GL(2,\ZZ)$ and $\PGL(2,\ZZ)$matrices with connections to cat maps and trace maps, J. Phys. A: Math. Gen., 30 (1997), 1549-1573. doi: 10.1088/0305-4470/30/5/020.
[10]	M. Baake and J. A . G. Roberts, symmetries of toral automorphisms, Nonlinearity, 14 (2001), R1-R24.
[11]	M. Baake and J. A. G. Roberts, The structure of reversing symmetry groups, Bull. Austral. Math. Soc., 73 (2006), 445-459.
[12]	M. Baake, J. A. G. Roberts and A. Weiss, Periodic orbits of linear endomorphisms of the $2$-torusand its lattices, Nonlinearity, 21 (2008), 2427-2446. doi: 10.1088/0951-7715/21/10/012.
[13]	E. Behrends and B. Fiedler, Periods of discretized linear Anosov maps, Ergod. Th. & Dynam. Syst., 18 (1998), 331-341. doi: 10.1017/S0143385798100366.
[14]	E. Brown and T. P. Vaughan, Cycles of directed graphs definedby matrix multiplication (mod $n$), Discr. Math., 239 (2001), 109-120. doi: 10.1016/S0012-365X(01)00040-1.
[15]	P. Bundschuh and J. S. Shiue, A generalization of a paper by D. D. Wall, Rendiconti Accademia Nazionale dei Lincei, Roma,Classe di Scienze Fisiche,Matematiche e Naturali, 56 (1974), 135-144.
[16]	D. Damanik, Gordon-type arguments in the spectral theory of one-dimensional quasicrystals, in "Directions in Mathematical Quasicrystals" (eds. M. Baake and R. V. Moody), CRM Monograph Series, AMS, Providence, RI, 13 (2000), 277-305.
[17]	R. W. Davis, Certain matrix equations over rings of integers, Duke Math. J., 35 (1968), 49-59. doi: 10.1215/S0012-7094-68-03506-0.
[18]	M. Degli Esposti and S. Isola, Distribution of closed orbits for linear automorphisms of tori, Nonlinearity, 8 (1995), 827-842. doi: 10.1088/0951-7715/8/5/010.
[19]	M. Degli Esposti and B. Winn, The quantum perturbed cat map and symmetry, J. Phys. A: Math. Gen., 38 (2005), 5895-5912. doi: 10.1088/0305-4470/38/26/005.
[20]	R. De Vogelaere, On the structure of symmetric periodic solutions of conservative systems, with applications, Ch. IV of Contributions to the Theory of NonlinearOscillations, ed. S. Lefschetz, Princeton Univ. Press, Princeton, IV 1958, 53-84.
[21]	F. J. Dyson and H. Falk, Period of a discrete cat mapping, Amer. Math. Monthly, 99 (1992), 603-614. doi: 10.2307/2324989.
[22]	H. T. Engstrom, On sequences defined by linear recurrence relations, Trans. Amer. Math. Soc., 33 (1931), 210-218. doi: 10.1090/S0002-9947-1931-1501585-5.
[23]	A. Fel'shtyn, Dynamical zeta functions, nielsen theory and reidemeister torsion, Memoirs AMS, Providence, RI, 147 (2000), xii+146 pp.
[24]	F. R. Gantmacher, "Matrix Theory," Chelsea, New York, I 1960.
[25]	G. Gaspari, The Arnold cat map on prime lattices, Physica, 73 (1994), 352-372.
[26]	H. Hasse, "Number Theory," Springer, Berlin 1980. doi: 10.1007/978-3-642-66671-1.
[27]	N. Jacobson, "Lectures in Abstract Algebra. II. Linear Algebra," reprint, Springer, New York, 1975.
[28]	A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995.
[29]	J. P. Keating, Asymptotic properties of the periodic orbits of the cat maps, Nonlinearity, 4 (1991), 277-307. doi: 10.1088/0951-7715/4/2/005.
[30]	J. P. Keating and F. Mezzadri, Pseudo-symmetries of Anosov maps and spectral statistics, Nonlinearity, 13 (2000), 747-775. doi: 10.1088/0951-7715/13/3/313.
[31]	P. Kurlberg, On the order of unimodular matrices modulo integers, Acta Arithm., 110 (2003), 141-151. doi: 10.4064/aa110-2-4.
[32]	P. Kurlberg and Z. Rudnick, Hecke theory and equidistribution for the quantization of linear maps of the torus, Duke Math. J., 103 (2000), 47-77. doi: 10.1215/S0012-7094-00-10314-6.
[33]	S. Lang, "Algebra," rev. 3rd ed., Springer, New York, 2002. doi: 10.1007/978-1-4613-0041-0.
[34]	R. Lidl and H. Niederreiter, "Introduction to Finite Fields and Their Applications," Cambridge University Press, Cambridge, 1986.
[35]	N. Neumärker, "Orbitstatistik und Relative Realisierbarkeit," Diploma Thesis, Univ. Bielefeld, 2007.
[36]	N. Neumärker, "The Arithmetic Structure of Discrete Dynamical Systems on the Torus," PhD thesis, Univ. Bielefeld, 2012.
[37]	I. Percival and F. Vivaldi, Arithmetical properties of strongly chaotic motions, Physica, 25 (1987), 105-130.
[38]	J. A. G. Roberts and M. Baake, Trace maps as 3D reversible dynamical systems with an invariant, J. Stat. Phys., 74 (1994), 829-888. doi: 10.1007/BF02188581.
[39]	J. A. G. Roberts and G. R. W. Quispel, Chaos and time-reversal symmetry. Order and chaos inreversible dynamical systems, Phys. Rep., 216 (1992), 63-177. doi: 10.1016/0370-1573(92)90163-T.
[40]	D. Ruelle, "Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval," CRM Monograph Series, AMS, Providence, RI, 4 1994.
[41]	P. Seibt, A period formula for torus automorphisms, Discr. Cont. Dynam. Syst., 9 (2003), 1029-1048. doi: 10.3934/dcds.2003.9.1029.
[42]	N. J. A. Sloane, "The Online Encyclopedia of Integer Sequences," http://oeis.org
[43]	O. Taussky, Introduction into connections between algebraic number theory and integral matrices, 2nd appendix to: H. Cohn, A Classical Invitation to Algebraic Numbers and Class Fields, 2nd printing, Springer,New York, (1988), 305-321.
[44]	D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532. doi: 10.2307/2309169.
[45]	P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.
[46]	M. Ward, The arithmetic theory of linear recurring sequences, Trans. Amer. Math. Soc., 35 (1933), 600-628. doi: 10.1090/S0002-9947-1933-1501705-4.
[47]	R. J. Wilson, "Introduction to Graph Theory," 4th ed., Prentice Hall, Harlow, 1996.
[48]	N. Zierler, Linear recurring sequences, J. Soc. Indust. Appl. Math., 7 (1959), 31-48. doi: 10.1137/0107003.

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