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