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A Brief Guide to Reversing and Extended Symmetries of Dynamical Systems

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Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2213))

Abstract

The reversing symmetry group is a well-studied extension of the symmetry group of a dynamical system, the latter being defined by the action of a single homeomorphism on a topological space. While it is traditionally considered in nonlinear dynamics, where the space is simple but the map is complicated, it has an interesting counterpart in symbolic dynamics, where the map is simple but the space is not. Moreover, there is an interesting extension to the case of higher-dimensional shifts, where a similar concept can be introduced via the centraliser and the normaliser of the acting group in the full automorphism group of the shift space. We recall the basic notions and review some of the known results, in a fairly informal manner, to give a first impression of the phenomena that can show up in the extension from the centraliser to the normaliser, with some emphasis on recent developments.

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References

  1. R.L. Adler, R. Palais, Homeomorphic conjugacy of automorphisms of the torus. Proc. Am. Math. Soc. 16, 1222–1225 (1965)

    Google Scholar 

  2. R.L. Adler, B. Weiss, Similarity of Automorphisms of the Torus. Memoirs AMS, vol. 98 (American Mathematical Society, Providence, RI, 1970)

    Google Scholar 

  3. J.-P. Allouche, J. Shallit, Automatic Sequences (Cambridge University Press, Cambridge, 2003)

    Google Scholar 

  4. T.M. Apostol, Introduction to Analytic Number Theory (Springer, New York, 1976)

    Google Scholar 

  5. V.I. Arnold, A. Avez, Ergodic Problems of Classical Mechanics (Addison-Wesley, Redwood City, CA, 1989)

    Google Scholar 

  6. M. Baake, Structure and representations of the hyperoctahedral group. J. Math. Phys. 25, 3171–3182 (1984)

    Google Scholar 

  7. M. Baake, U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation (Cambridge University Press, Cambridge, 2013)

    Google Scholar 

  8. M. Baake, U. Grimm, Squirals and beyond: substitution tilings with singular continuous spectrum. Ergodic Theory Dyn. Syst. 34, 1077–1102 (2014). arXiv:1205.1384

    Google Scholar 

  9. M. Baake, J.A.G. Roberts, Reversing symmetry group of GL(2, Z) and PGL(2, Z) matrices with connections to cat maps and trace maps. J. Phys. A: Math. Gen. 30, 1549–1573 (1997)

    Google Scholar 

  10. M. Baake, J.A.G. Roberts, Symmetries and reversing symmetries of toral automorphisms. Nonlinearity 14, R1–R24 (2001). arXiv:math.DS/0006092

    Google Scholar 

  11. M. Baake, J.A.G. Roberts, Symmetries and reversing symmetries of polynomial automorphisms of the plane. Nonlinearity 18, 791–816 (2005). arXiv:math.DS/0501151

    Google Scholar 

  12. M. Baake, J.A.G. Roberts, The structure of reversing symmetry groups. Bull. Aust. Math. Soc. 73, 445–459 (2006). arXiv:math.DS/0605296

    Google Scholar 

  13. M. Baake, T. Ward, Planar dynamical systems with pure Lebesgue diffraction spectrum. J. Stat. Phys. 140, 90–102 (2010). arXiv:1003.1536

    Google Scholar 

  14. M. Baake, U. Grimm, D. Joseph, Trace maps, invariants, and some of their applications. Int. J. Mod. Phys. B 7, 1527–1550 (1993). arXiv:math-ph/9904025

    Google Scholar 

  15. M. Baake, R.V. Moody, P.A.B. Pleasants, Diffraction from visible lattice points and k-th power free integers. Discrete Math. 221, 3–42 (2000). arXiv:math.MG/9906132

    Google Scholar 

  16. M. Baake, J.A.G. Roberts, A. Weiss, Periodic orbits of linear endomorphisms on the 2-torus and its lattices. Nonlinearity 21, 2427–2446 (2008). arXiv:0808.3489

    Google Scholar 

  17. M. Baake, F. Gähler, U. Grimm, Spectral and topological properties of a family of generalised Thue–Morse sequences. J. Math. Phys. 53, 032701:1–24 (2012). arXiv:1201.1423

    Google Scholar 

  18. M. Baake, N. Neumärker, J.A.G. Roberts, Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices. Discrete Contin. Dyn. Syst. A 33, 527–553 (2013). arXiv:1205.1003

    Google Scholar 

  19. M. Baake, C. Huck, N. Strungaru, On weak model sets of extremal density. Ind. Math. 28, 3–31 (2017). arXiv:1512.07129

    Google Scholar 

  20. M. Baake, J.A.G. Roberts, R. Yassawi, Reversing and extended symmetries of shift spaces. Discrete Contin. Dyn. Syst. A 38, 835–866 (2018). arXiv:1611.05756

    Google Scholar 

  21. M. Baake, C. Huck, M. Lemańczyk, Positive entropy shifts with small centraliser and large normaliser (in preparation)

    Google Scholar 

  22. A. Bartnicka, Automorphisms of Toeplitz \({\mathbb {B}}\)-free systems. Preprint. arXiv:1705.07021

    Google Scholar 

  23. S. Bhattacharya, K. Schmidt, Homoclinic points and isomorphism rigidity of algebraic \({\mathbb {Z}}^{d}\)-actions on zero-dimensional compact Abelian groups. Isr. J. Math. 137, 189–209 (2003)

    Google Scholar 

  24. Z.I. Borevich, I.R. Shafarevich, Number Theory (Academic Press, New York, 1966)

    Google Scholar 

  25. W. Bulatek, J. Kwiatkowski, Strictly ergodic Toeplitz flows with positive entropy and trivial centralizers. Stud. Math. 103, 133–142 (1992)

    Google Scholar 

  26. E.M. Coven, Endomorphisms of substitution minimal sets. Z. Wahrscheinlichkeitsth. verw. Geb. 20, 129–133 (1971/1972)

    Google Scholar 

  27. E.M. Coven, G.A. Hedlund, Sequences with minimal block growth. Math. Syst. Theory 7, 138–153 (1973)

    Google Scholar 

  28. E.M. Coven, A. Quas, R. Yassawi, Computing automorphism groups of shifts using atypical equivalence classes. Discrete Anal. 2016(3) (28pp). arXiv:1505.02482

    Google Scholar 

  29. V. Cyr, B. Kra, The automorphism group of a shift of linear growth: beyond transitivity. Forum Math. Sigma 3, e5 27 (2015). arXiv:1411.0180

    Google Scholar 

  30. V. Cyr, B. Kra, The automorphism group of a shift of subquadratic growth. Proc. Am. Math. Soc. 2, 613–621 (2016). arXiv:1403.0238

    Google Scholar 

  31. D. Damanik, A. Gorodetski, W. Yessen, The Fibonacci Hamiltonian. Invent. Math. 206, 629–692 (2016). arXiv:1403.7823

    Google Scholar 

  32. S. Donoso, F. Durand, A. Maass, S. Petite, On automorphism groups of low complexity shifts. Ergodic Theory Dyn. Syst. 36, 64–95 (2016). arXiv:1501.0051

    Google Scholar 

  33. X. Droubay, G. Pirillo, Palindromes and Sturmian words. Theor. Comput. Sci. 223, 73–85 (1999)

    Google Scholar 

  34. E.H. El Abdalaoui, M. Lemańczyk, T. de la Rue, A dynamical point of view on the set of \({\mathbb {B}}\)-free integers. Int. Math. Res. Not. 2015(16), 7258–7286 (2015). arXiv:1311.3752

    Google Scholar 

  35. K. Fra̧czek, J. Kułaga, M. Lemańczyk, Non-reversibility and self-joinings of higher orders for ergodic flows. J. Anal. Math. 122, 163–227 (2014). arXiv:1206.3053

    Google Scholar 

  36. N.P. Frank, Multi-dimensional constant-length substitution sequences. Topology Appl. 152, 44–69 (2005)

    Google Scholar 

  37. A. Gómez, J. Meiss, Reversors and symmetries for polynomial automorphisms of the complex plane. Nonlinearity 17, 975–1000 (2004). arXiv:nlin.CD/0304035

    Google Scholar 

  38. G.R. Goodson, Inverse conjugacies and reversing symmetry groups. Am. Math. Mon. 106, 19–26 (1999)

    Google Scholar 

  39. G. Goodson, A. del Junco, M. Lemańczyk, D. Rudolph, Ergodic transformation conjugate to their inverses by involutions. Ergodic Theory Dyn. Syst. 16, 97–124 (1996)

    Google Scholar 

  40. M. Hochman, Genericity in topological dynamics. Ergodic Theory Dyn. Syst. 28, 125–165 (2008)

    Google Scholar 

  41. N. Jacobson, Lectures in Abstract Algebra. II. Linear Algebra (Springer, New York, 1953)

    Google Scholar 

  42. H.W.E. Jung, Über ganze irrationale Transformationen der Ebene. J. Reine Angew. Math. (Crelle) 184, 161–174 (1942)

    Google Scholar 

  43. A. Karrass, D. Solitar, The subgroups of a free product of two groups with an amalgamated subgroup. Trans. Am. Math. Soc. 150, 227–255 (1970)

    Google Scholar 

  44. S. Kasjan, G. Keller, M. Lemańczyk, Dynamics of \({\mathbb {B}}\)-free sets: a view through the window. Int. Math. Res. Not. (2017, in press). arXiv:1702.02375

    Google Scholar 

  45. M. Keane, Generalized Morse sequences. Z. Wahrscheinlichkeitsth. verw. Geb. 10, 335–353 (1968)

    Google Scholar 

  46. G. Keller, Generalized heredity in \({\mathbb {B}}\)-free systems. Preprint. arXiv:1704.04079

    Google Scholar 

  47. G. Keller, C. Richard, Dynamics on the graph of the torus parametrisation. Ergodic Theory Dyn. Syst. (in press). arXiv:1511.06137

    Google Scholar 

  48. G. Keller, C. Richard, Periods and factors of weak model sets. Preprint. arXiv:1702.02383

    Google Scholar 

  49. Y.-O. Kim, J. Lee, K.K. Park, A zeta function for flip systems. Pac. J. Math. 209, 289–301 (2003)

    Google Scholar 

  50. B.P. Kitchens, Symbolic Dynamics (Springer, Berlin, 1998)

    Google Scholar 

  51. B. Kitchens, K. Schmidt, Isomorphism rigidity of irreducible algebraic \({\mathbb {Z}}^{d}\)-actions. Invent. Math. 142, 559–577 (2000)

    Google Scholar 

  52. J.S.W. Lamb, Reversing symmetries in dynamical systems. J. Phys. A: Math. Gen. 25, 925–937 (1992)

    Google Scholar 

  53. J.S.W. Lamb, J.A.G. Roberts, Time-reversal symmetry in dynamical systems: a survey. Physica D 112, 1–39 (1998)

    Google Scholar 

  54. F. Ledrappier, Un champ markovien peut être d’entropie nulle et mélangeant. C. R. Acad. Sci. Paris Sér. A-B 287, A561–A563 (1978)

    Google Scholar 

  55. J. Lee, K.K. Park, S. Shin, Reversible topological Markov shifts. Ergodic Theory Dyn. Syst. 26, 267–280 (2006)

    Google Scholar 

  56. D.A. Lind, B. Marcus, An Introduction to Symbolic Dynamics and Coding (Cambridge University Press, Cambridge, 1995)

    Google Scholar 

  57. M.K. Mentzen, Automorphisms of shifts defined by \(\mathbb {B}\)-free sets of integers. Coll. Math. 147, 87–94 (2017)

    Google Scholar 

  58. Y. Meyer, Algebraic Numbers and Harmonic Analysis (North Holland, Amsterdam, 1972)

    Google Scholar 

  59. M. Morse, G.A. Hedlund, Symbolic dynamics II. Sturmian trajectories. Am. J. Math. 62, 1–42 (1940)

    Google Scholar 

  60. A.G. O’Farrel, I. Short, Reversibility in Dynamics and Group Theory (Cambridge University Press, Cambridge, 2015)

    Google Scholar 

  61. J. Olli, Endomorphisms of Sturmian systems and the discrete chair substitution tiling system. Discrete Contin. Dyn. Syst. A 33, 4173–4186 (2013)

    Google Scholar 

  62. K. Petersen, Ergodic Theory (Cambridge University Press, Cambridge, 1983)

    Google Scholar 

  63. N. Pytheas Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics. Lecture Notes in Mathematics, vol. 1794 (Springer, Berlin, 2002)

    Google Scholar 

  64. M. Queffélec, Substitution Dynamical Systems – Spectral Analysis. Lecture Notes in Mathematics, vol. 1294, 2nd edn. (Springer, Berlin, 2010)

    Google Scholar 

  65. J.A.G. Roberts, M. Baake, Trace maps as 3D reversible dynamical systems with an invariant. J. Stat. Phys. 74, 829–888 (1994)

    Google Scholar 

  66. J.A.G. Roberts, M. Baake, Symmetries and reversing symmetries of area-preserving polynomial mappings in generalised standard form. Physica A 317, 95–112 (2002). arXiv:math.DS/0206096

    Google Scholar 

  67. E.A. Robinson, On the table and the chair. Indag. Math. 10, 581–599 (1999)

    Google Scholar 

  68. J.-P. Serre, Trees, 2nd. corr. printing (Springer, Berlin, 2003)

    Google Scholar 

  69. K. Schmidt, Dynamical Systems of Algebraic Origin (Birkhäuser, Basel, 1995)

    Google Scholar 

  70. R.L.E. Schwarzenberger, N-Dimensional Crystallography (Pitman, San Francisco, 1980)

    MATH  Google Scholar 

  71. M.B. Sevryuk, Reversible Systems. Lecture Notes in Mathematics, vol. 1211 (Springer, Berlin, 1986)

    Google Scholar 

  72. W. van der Kulk, On polynomial rings in two variables. Nieuw Arch. Wisk. 1, 33–41 (1953)

    MathSciNet  MATH  Google Scholar 

  73. D. Wright, Abelian subgroups of Aut k (k[X, Y ]) and applications to actions on the affine plane. Ill. J. Math. 23, 579–634 (1979)

    MATH  Google Scholar 

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Acknowledgements

A substantial part of this exposition is based on a joint work (both past and ongoing) with John Roberts, who introduced me to the concepts over 25 years ago. More recent activities also profited a lot from the interaction and cooperation with Christian Huck, Mariusz Lemańczyk and Reem Yassawi. It is a pleasure to thank the CIRM in Luminy for its support and the stimulating atmosphere during the special program in the framework of the Jean Morlet semester ‘Ergodic Theory and Dynamical Systems in their Interactions with Arithmetic and Combinatorics’, where part of this work was done.

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  1. Faculty of Mathematics, Bielefeld University, Bielefeld, Germany

    M. Baake

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  1. M. Baake
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Correspondence to M. Baake .

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Editors and Affiliations

  1. CNRS UMR 7373, Institut de Mathématiques de Marseille, Marseille, France

    Sébastien Ferenczi

  2. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland

    Joanna Kułaga-Przymus

  3. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland

    Mariusz Lemańczyk

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Baake, M. (2018). A Brief Guide to Reversing and Extended Symmetries of Dynamical Systems. In: Ferenczi, S., Kułaga-Przymus, J., Lemańczyk, M. (eds) Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics. Lecture Notes in Mathematics, vol 2213. Springer, Cham. https://doi.org/10.1007/978-3-319-74908-2_9

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