# American Institute of Mathematical Sciences

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

## 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

Received  August 2011 Revised  May 2012 Published  September 2012

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.
Citation: Michael Baake, Natascha Neumärker, John A. G. Roberts. Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices. Discrete & Continuous Dynamical Systems - A, 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, (1999). Google Scholar [2] R. Adler, C. Tresser and P. A. Worfolk, Topological conjugacy of linear endomorphisms of the 2-torus,, Trans. AMS, 349 (1997), 1633. doi: 10.1090/S0002-9947-97-01895-3. Google Scholar [3] V. I. Arnold and A. Avez, "Ergodic Problems of Classical Mechanics,", reprint, (1989). Google Scholar [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. doi: 10.1080/00927870902747266. Google Scholar [5] H. Aydin, R. Dikici and G. C. Smith, Wall and Vinston revisited,, in, 5 (1992), 61. Google Scholar [6] M. Baake, U. Grimm and D. Joseph, Trace maps, invariants, and some of their applications,, Int. J. Mod. Phys. B, 7 (1993), 1527. doi: 10.1142/S021797929300247X. Google Scholar [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. Google Scholar [8] M. Baake, E. Lau and V. Paskunas, A note on the dynamical zeta function of general toral endomorphisms,, Monatsh. Math., 161 (2010), 33. Google Scholar [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. doi: 10.1088/0305-4470/30/5/020. Google Scholar [10] M. Baake and J. A . G. Roberts, symmetries of toral automorphisms,, Nonlinearity, 14 (2001). Google Scholar [11] M. Baake and J. A. G. Roberts, The structure of reversing symmetry groups,, Bull. Austral. Math. Soc., 73 (2006), 445. Google Scholar [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. doi: 10.1088/0951-7715/21/10/012. Google Scholar [13] E. Behrends and B. Fiedler, Periods of discretized linear Anosov maps,, Ergod. Th. & Dynam. Syst., 18 (1998), 331. doi: 10.1017/S0143385798100366. Google Scholar [14] E. Brown and T. P. Vaughan, Cycles of directed graphs definedby matrix multiplication (mod $n$),, Discr. Math., 239 (2001), 109. doi: 10.1016/S0012-365X(01)00040-1. Google Scholar [15] P. Bundschuh and J. S. Shiue, A generalization of a paper by D. D. Wall,, Rendiconti Accademia Nazionale dei Lincei, 56 (1974), 135. Google Scholar [16] D. Damanik, Gordon-type arguments in the spectral theory of one-dimensional quasicrystals,, in, 13 (2000), 277. Google Scholar [17] R. W. Davis, Certain matrix equations over rings of integers,, Duke Math. J., 35 (1968), 49. doi: 10.1215/S0012-7094-68-03506-0. Google Scholar [18] M. Degli Esposti and S. Isola, Distribution of closed orbits for linear automorphisms of tori,, Nonlinearity, 8 (1995), 827. doi: 10.1088/0951-7715/8/5/010. Google Scholar [19] M. Degli Esposti and B. Winn, The quantum perturbed cat map and symmetry,, J. Phys. A: Math. Gen., 38 (2005), 5895. doi: 10.1088/0305-4470/38/26/005. Google Scholar [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, IV (1958), 53. Google Scholar [21] F. J. Dyson and H. Falk, Period of a discrete cat mapping,, Amer. Math. Monthly, 99 (1992), 603. doi: 10.2307/2324989. Google Scholar [22] H. T. Engstrom, On sequences defined by linear recurrence relations,, Trans. Amer. Math. Soc., 33 (1931), 210. doi: 10.1090/S0002-9947-1931-1501585-5. Google Scholar [23] A. Fel'shtyn, Dynamical zeta functions, nielsen theory and reidemeister torsion,, Memoirs AMS, 147 (2000). Google Scholar [24] F. R. Gantmacher, "Matrix Theory,", Chelsea, I (1960). Google Scholar [25] G. Gaspari, The Arnold cat map on prime lattices,, Physica, 73 (1994), 352. Google Scholar [26] H. Hasse, "Number Theory,", Springer, (1980). doi: 10.1007/978-3-642-66671-1. Google Scholar [27] N. Jacobson, "Lectures in Abstract Algebra. II. Linear Algebra,", reprint, (1975). Google Scholar [28] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge University Press, (1995). Google Scholar [29] J. P. Keating, Asymptotic properties of the periodic orbits of the cat maps,, Nonlinearity, 4 (1991), 277. doi: 10.1088/0951-7715/4/2/005. Google Scholar [30] J. P. Keating and F. Mezzadri, Pseudo-symmetries of Anosov maps and spectral statistics,, Nonlinearity, 13 (2000), 747. doi: 10.1088/0951-7715/13/3/313. Google Scholar [31] P. Kurlberg, On the order of unimodular matrices modulo integers,, Acta Arithm., 110 (2003), 141. doi: 10.4064/aa110-2-4. Google Scholar [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. doi: 10.1215/S0012-7094-00-10314-6. Google Scholar [33] S. Lang, "Algebra,", rev. 3rd ed., (2002). doi: 10.1007/978-1-4613-0041-0. Google Scholar [34] R. Lidl and H. Niederreiter, "Introduction to Finite Fields and Their Applications,", Cambridge University Press, (1986). Google Scholar [35] N. Neumärker, "Orbitstatistik und Relative Realisierbarkeit,", Diploma Thesis, (2007). Google Scholar [36] N. Neumärker, "The Arithmetic Structure of Discrete Dynamical Systems on the Torus,", PhD thesis, (2012). Google Scholar [37] I. Percival and F. Vivaldi, Arithmetical properties of strongly chaotic motions,, Physica, 25 (1987), 105. Google Scholar [38] J. A. G. Roberts and M. Baake, Trace maps as 3D reversible dynamical systems with an invariant,, J. Stat. Phys., 74 (1994), 829. doi: 10.1007/BF02188581. Google Scholar [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. doi: 10.1016/0370-1573(92)90163-T. Google Scholar [40] D. Ruelle, "Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval,", CRM Monograph Series, 4 (1994). Google Scholar [41] P. Seibt, A period formula for torus automorphisms,, Discr. Cont. Dynam. Syst., 9 (2003), 1029. doi: 10.3934/dcds.2003.9.1029. Google Scholar [42] N. J. A. Sloane, "The Online Encyclopedia of Integer Sequences,", , (). Google Scholar [43] O. Taussky, Introduction into connections between algebraic number theory and integral matrices,, 2nd appendix to: H. Cohn, (1988), 305. Google Scholar [44] D. D. Wall, Fibonacci series modulo m,, Amer. Math. Monthly, 67 (1960), 525. doi: 10.2307/2309169. Google Scholar [45] P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, (1982). Google Scholar [46] M. Ward, The arithmetic theory of linear recurring sequences,, Trans. Amer. Math. Soc., 35 (1933), 600. doi: 10.1090/S0002-9947-1933-1501705-4. Google Scholar [47] R. J. Wilson, "Introduction to Graph Theory,", 4th ed., (1996). Google Scholar [48] N. Zierler, Linear recurring sequences,, J. Soc. Indust. Appl. Math., 7 (1959), 31. doi: 10.1137/0107003. Google Scholar

show all references

##### References:
 [1] W. A. Adkins and S. H. Weintraub, "Algebra - An Approach via Module Theory,", corr. 2nd printing, (1999). Google Scholar [2] R. Adler, C. Tresser and P. A. Worfolk, Topological conjugacy of linear endomorphisms of the 2-torus,, Trans. AMS, 349 (1997), 1633. doi: 10.1090/S0002-9947-97-01895-3. Google Scholar [3] V. I. Arnold and A. Avez, "Ergodic Problems of Classical Mechanics,", reprint, (1989). Google Scholar [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. doi: 10.1080/00927870902747266. Google Scholar [5] H. Aydin, R. Dikici and G. C. Smith, Wall and Vinston revisited,, in, 5 (1992), 61. Google Scholar [6] M. Baake, U. Grimm and D. Joseph, Trace maps, invariants, and some of their applications,, Int. J. Mod. Phys. B, 7 (1993), 1527. doi: 10.1142/S021797929300247X. Google Scholar [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. Google Scholar [8] M. Baake, E. Lau and V. Paskunas, A note on the dynamical zeta function of general toral endomorphisms,, Monatsh. Math., 161 (2010), 33. Google Scholar [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. doi: 10.1088/0305-4470/30/5/020. Google Scholar [10] M. Baake and J. A . G. Roberts, symmetries of toral automorphisms,, Nonlinearity, 14 (2001). Google Scholar [11] M. Baake and J. A. G. Roberts, The structure of reversing symmetry groups,, Bull. Austral. Math. Soc., 73 (2006), 445. Google Scholar [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. doi: 10.1088/0951-7715/21/10/012. Google Scholar [13] E. Behrends and B. Fiedler, Periods of discretized linear Anosov maps,, Ergod. Th. & Dynam. Syst., 18 (1998), 331. doi: 10.1017/S0143385798100366. Google Scholar [14] E. Brown and T. P. Vaughan, Cycles of directed graphs definedby matrix multiplication (mod $n$),, Discr. Math., 239 (2001), 109. doi: 10.1016/S0012-365X(01)00040-1. Google Scholar [15] P. Bundschuh and J. S. Shiue, A generalization of a paper by D. D. Wall,, Rendiconti Accademia Nazionale dei Lincei, 56 (1974), 135. Google Scholar [16] D. Damanik, Gordon-type arguments in the spectral theory of one-dimensional quasicrystals,, in, 13 (2000), 277. Google Scholar [17] R. W. Davis, Certain matrix equations over rings of integers,, Duke Math. J., 35 (1968), 49. doi: 10.1215/S0012-7094-68-03506-0. Google Scholar [18] M. Degli Esposti and S. Isola, Distribution of closed orbits for linear automorphisms of tori,, Nonlinearity, 8 (1995), 827. doi: 10.1088/0951-7715/8/5/010. Google Scholar [19] M. Degli Esposti and B. Winn, The quantum perturbed cat map and symmetry,, J. Phys. A: Math. Gen., 38 (2005), 5895. doi: 10.1088/0305-4470/38/26/005. Google Scholar [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, IV (1958), 53. Google Scholar [21] F. J. Dyson and H. Falk, Period of a discrete cat mapping,, Amer. Math. Monthly, 99 (1992), 603. doi: 10.2307/2324989. Google Scholar [22] H. T. Engstrom, On sequences defined by linear recurrence relations,, Trans. Amer. Math. Soc., 33 (1931), 210. doi: 10.1090/S0002-9947-1931-1501585-5. Google Scholar [23] A. Fel'shtyn, Dynamical zeta functions, nielsen theory and reidemeister torsion,, Memoirs AMS, 147 (2000). Google Scholar [24] F. R. Gantmacher, "Matrix Theory,", Chelsea, I (1960). Google Scholar [25] G. Gaspari, The Arnold cat map on prime lattices,, Physica, 73 (1994), 352. Google Scholar [26] H. Hasse, "Number Theory,", Springer, (1980). doi: 10.1007/978-3-642-66671-1. Google Scholar [27] N. Jacobson, "Lectures in Abstract Algebra. II. Linear Algebra,", reprint, (1975). Google Scholar [28] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge University Press, (1995). Google Scholar [29] J. P. Keating, Asymptotic properties of the periodic orbits of the cat maps,, Nonlinearity, 4 (1991), 277. doi: 10.1088/0951-7715/4/2/005. Google Scholar [30] J. P. Keating and F. Mezzadri, Pseudo-symmetries of Anosov maps and spectral statistics,, Nonlinearity, 13 (2000), 747. doi: 10.1088/0951-7715/13/3/313. Google Scholar [31] P. Kurlberg, On the order of unimodular matrices modulo integers,, Acta Arithm., 110 (2003), 141. doi: 10.4064/aa110-2-4. Google Scholar [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. doi: 10.1215/S0012-7094-00-10314-6. Google Scholar [33] S. Lang, "Algebra,", rev. 3rd ed., (2002). doi: 10.1007/978-1-4613-0041-0. Google Scholar [34] R. Lidl and H. Niederreiter, "Introduction to Finite Fields and Their Applications,", Cambridge University Press, (1986). Google Scholar [35] N. Neumärker, "Orbitstatistik und Relative Realisierbarkeit,", Diploma Thesis, (2007). Google Scholar [36] N. Neumärker, "The Arithmetic Structure of Discrete Dynamical Systems on the Torus,", PhD thesis, (2012). Google Scholar [37] I. Percival and F. Vivaldi, Arithmetical properties of strongly chaotic motions,, Physica, 25 (1987), 105. Google Scholar [38] J. A. G. Roberts and M. Baake, Trace maps as 3D reversible dynamical systems with an invariant,, J. Stat. Phys., 74 (1994), 829. doi: 10.1007/BF02188581. Google Scholar [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. doi: 10.1016/0370-1573(92)90163-T. Google Scholar [40] D. Ruelle, "Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval,", CRM Monograph Series, 4 (1994). Google Scholar [41] P. Seibt, A period formula for torus automorphisms,, Discr. Cont. Dynam. Syst., 9 (2003), 1029. doi: 10.3934/dcds.2003.9.1029. Google Scholar [42] N. J. A. Sloane, "The Online Encyclopedia of Integer Sequences,", , (). Google Scholar [43] O. Taussky, Introduction into connections between algebraic number theory and integral matrices,, 2nd appendix to: H. Cohn, (1988), 305. Google Scholar [44] D. D. Wall, Fibonacci series modulo m,, Amer. Math. Monthly, 67 (1960), 525. doi: 10.2307/2309169. Google Scholar [45] P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, (1982). Google Scholar [46] M. Ward, The arithmetic theory of linear recurring sequences,, Trans. Amer. Math. Soc., 35 (1933), 600. doi: 10.1090/S0002-9947-1933-1501705-4. Google Scholar [47] R. J. Wilson, "Introduction to Graph Theory,", 4th ed., (1996). Google Scholar [48] N. Zierler, Linear recurring sequences,, J. Soc. Indust. Appl. Math., 7 (1959), 31. doi: 10.1137/0107003. Google Scholar
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