Figure 1.
The shifted lattice $ p+ \mathbb{Z} $ is coded as a bi-infinite binary sequence $ w\in\{L, R\}^ \mathbb{Z} $ which is a symbolic representation of $ p $
-
Previous Article
On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł, and Tao
- JMD Home
- This Volume
-
Next Article
Dynamics of transcendental Hénon maps III: Infinite entropy
2021, 17: 481-528.
doi: 10.3934/jmd.2021017
Rauzy induction of polygon partitions and toral $ \mathbb{Z}^2 $-rotations
Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400, Talence, France |
We extend the notion of Rauzy induction of interval exchange transformations to the case of toral $ \mathbb{Z}^2 $-rotation, i.e., $ \mathbb{Z}^2 $-action defined by rotations on a 2-torus. If $ \mathscr{X}_{\mathscr{P}, R} $ denotes the symbolic dynamical system corresponding to a partition $ \mathscr{P} $ and $ \mathbb{Z}^2 $-action $ R $ such that $ R $ is Cartesian on a sub-domain $ W $, we express the 2-dimensional configurations in $ \mathscr{X}_{\mathscr{P}, R} $ as the image under a $ 2 $-dimensional morphism (up to a shift) of a configuration in $ \mathscr{X}_{\widehat{\mathscr{P}}|_W, \widehat{R}|_W} $ where $ \widehat{\mathscr{P}}|_W $ is the induced partition and $ \widehat{R}|_W $ is the induced $ \mathbb{Z}^2 $-action on $ W $.
We focus on one example, $ \mathscr{X}_{\mathscr{P}_0, R_0} $, for which we obtain an eventually periodic sequence of 2-dimensional morphisms. We prove that it is the same as the substitutive structure of the minimal subshift $ X_0 $ of the Jeandel–Rao Wang shift computed in an earlier work by the author. As a consequence, $ {\mathscr{P}}_0 $ is a Markov partition for the associated toral $ \mathbb{Z}^2 $-rotation $ R_0 $. It also implies that the subshift $ X_0 $ is uniquely ergodic and is isomorphic to the toral $ \mathbb{Z}^2 $-rotation $ R_0 $ which can be seen as a generalization for 2-dimensional subshifts of the relation between Sturmian sequences and irrational rotations on a circle. Batteries included: the algorithms and code to reproduce the proofs are provided.
Keywords:
Rauzy induction,
Markov partition,
self-induced,
self-similar,
SFT,
polyhedron exchange transformation,
aperiodic tiling,
Sturmian.
Mathematics Subject Classification:
37A05.
Citation:
Sébastien Labbé. Rauzy induction of polygon partitions and toral $ \mathbb{Z}^2 $-rotations. Journal of Modern Dynamics,
2021, 17: 481-528.
doi: 10.3934/jmd.2021017
![]()
References:
| [1] |
R. L. Adler,
Symbolic dynamics and Markov partitions, Bull. Amer. Math. Soc. (N.S.), 35 (1998), 1-56.
doi: 10.1090/S0273-0979-98-00737-X. |
| [2] |
R. L. Adler and B. Weiss, Similarity of Automorphisms of the Torus, Memoirs of the American Mathematical Society, 98, American Mathematical Society, Providence, R.I., 1970. |
| [3] |
S. Akiyama, M. Barge, V. Berthé, J.-Y. Lee and A. Siegel, On the Pisot substitution conjecture, in Mathematics of Aperiodic Order, Progr. Math., 309, Birkhäuser/Springer, Basel, 2015, 33–72.
doi: 10.1007/978-3-0348-0903-0_2. |
| [4] |
I. Alevy, R. Kenyon and R. Yi,
A family of minimal and renormalizable rectangle exchange maps, Ergodic Theory Dynam. Systems, 41 (2021), 790-817.
doi: 10.1017/etds.2019.77. |
| [5] |
P. Arnoux, Sturmian sequences, in Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Math., 1794, Springer, Berlin, 2002,143–198.
doi: 10.1007/3-540-45714-3_6. |
| [6] |
P. Arnoux, V. Berthé and S. Ito,
Discrete planes, $\Bbb Z^2$-actions, Jacobi-Perron algorithm and substitutions, Ann. Inst. Fourier (Grenoble), 52 (2002), 305-349.
doi: 10.5802/aif.1889. |
| [7] |
P. Arnoux, V. Berthé, H. Ei and S. Ito, Tilings, quasicrystals, discrete planes, generalized substitutions, and multidimensional continued fractions, in Discrete Models: Combinatorics, Computation, and Geometry (Paris, 2001), Discrete Math. Theor. Comput. Sci. Proc., AA, Maison Inform. Math. Discrèt. (MIMD), Paris, 2001, 59–78. |
| [8] |
A. Avila and G. Forni,
Weak mixing for interval exchange transformations and translation flows, Ann. of Math. (2), 165 (2007), 637-664.
doi: 10.4007/annals.2007.165.637. |
| [9] |
M. Baake, J. A. G. Roberts and R. Yassawi,
Reversing and extended symmetries of shift spaces, Discrete Contin. Dyn. Syst., 38 (2018), 835-866.
doi: 10.3934/dcds.2018036. |
| [10] |
R. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc., 66 (1966), 72 pp. |
| [11] |
V. Berthé, Arithmetic discrete planes are quasicrystals, in Discrete Geometry for Computer Imagery, Lecture Notes in Comput. Sci., 5810, Springer, Berlin, 2009, 1–12.
doi: 10.1007/978-3-642-04397-0_1. |
| [12] |
V. Berthé, J. Bourdon, T. Jolivet and A. Siegel, Generating discrete planes with substitutions, in Combinatorics on Words, Lecture Notes in Comput. Sci., 8079, Springer, Heidelberg, 2013, 58–70.
doi: 10.1007/978-3-642-40579-2_9. |
| [13] |
V. Berthé and V. Delecroix, Beyond substitutive dynamical systems: $S$-adic expansions, in Numeration and Substitution 2012, RIMS Kôkyûroku Bessatsu, B46, Res. Inst. Math. Sci. (RIMS), Kyoto, 2014, 81–123. |
| [14] |
V. Berthé, S. Ferenczi, and L. Q. Zamboni, Interactions between dynamics, arithmetics and combinatorics: The good, the bad, and the ugly, in Algebraic and Topological Dynamics, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005,333–364.
doi: 10.1090/conm/385. |
| [15] |
V. Berthé and M. Rigo, editors, Combinatorics, automata and number theory, Encyclopedia of Mathematics and its Applications, 135, Cambridge University Press, Cambridge, 2010.
doi: 10.1017/CBO9780511777653. |
| [16] |
V. Berthé, W. Steiner and J. M. Thuswaldner,
Geometry, dynamics, and arithmetic of $S$-adic shifts, Ann. Inst. Fourier (Grenoble), 69 (2019), 1347-1409.
doi: 10.5802/aif.3273. |
| [17] |
R. Bowen,
Markov partitions for Axiom A diffeomorphisms, Amer. J. Math., 92 (1970), 725-747.
doi: 10.2307/2373370. |
| [18] |
R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. |
| [19] |
R. Bowen,
Markov partitions are not smooth, Proc. Amer. Math. Soc., 71 (1978), 130-132.
doi: 10.2307/2042234. |
| [20] |
M. Boyle and J. Tomiyama,
Bounded topological orbit equivalence and $C^*$-algebras, J. Math. Soc. Japan, 50 (1998), 317-329.
doi: 10.2969/jmsj/05020317. |
| [21] |
V. Brun, Algorithmes euclidiens pour trois et quatre nombres, in Treizième Congrès des Mathématiciens Scandinaves, tenu à Helsinki 18-23 aoşt 1957, Mercators Tryckeri, Helsinki, 1958, 45–64. |
| [22] |
E. Cawley,
Smooth Markov partitions and toral automorphisms, Ergodic Theory Dynam. Systems, 11 (1991), 633-651.
doi: 10.1017/S0143385700006404. |
| [23] |
E. Charlier, T. Kärki and M. Rigo,
Multidimensional generalized automatic sequences and shape-symmetric morphic words, Discrete Math., 310 (2010), 1238-1252.
doi: 10.1016/j.disc.2009.12.002. |
| [24] |
E. M. Coven and G. A. Hedlund,
Sequences with minimal block growth, Math. Systems Theory, 7 (1973), 138-153.
doi: 10.1007/BF01762232. |
| [25] |
M. Einsiedler and K. Schmidt,
Markov partitions and homoclinic points of algebraic $\mathbf{Z}^d$-actions, Tr. Mat. Inst. Steklova, 216 (1997), 265-284.
|
| [26] |
N. P. Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, 1794, Springer-Verlag, Berlin, 2002., Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel.
doi: 10.1007/b13861. |
| [27] |
M. Hochman, Multidimensional shifts of finite type and sofic shifts, in Combinatorics, Words
and Symbolic Dynamics, Encyclopedia Math. Appl., 159, Cambridge Univ. Press, Cambridge,
2016,296–358.
doi: 10.1017/CBO9781139924733.010. |
| [28] |
M. Hochman and T. Meyerovitch,
A characterization of the entropies of multidimensional shifts of finite type, Ann. of Math. (2), 171 (2010), 2011-2038.
doi: 10.4007/annals.2010.171.2011. |
| [29] |
W. P. Hooper,
Renormalization of polygon exchange maps arising from corner percolation, Invent. Math., 191 (2013), 255-320.
doi: 10.1007/s00222-012-0393-4. |
| [30] |
E. Jeandel and M. Rao, An aperiodic set of 11 Wang tiles, Adv. Comb., (2021), 37 pp.
doi: 10.19086/aic.18614. |
| [31] |
R. Kenyon and A. Vershik,
Arithmetic construction of sofic partitions of hyperbolic toral automorphisms, Ergodic Theory Dynam. Systems, 18 (1998), 357-372.
doi: 10.1017/S0143385798100445. |
| [32] |
B. Kitchens, Symbolic dynamics, group automorphisms and Markov partitions, in Real and Complex Dynamical Systems (Hillerød, 1993), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995,133–163. |
| [33] |
B. P. Kitchens, Symbolic Dynamics. One-sided, two-sided and countable state Markov shifts, Universitext, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-642-58822-8. |
| [34] |
S. Labbé,
A self-similar aperiodic set of 19 Wang tiles, Geom. Dedicata, 201 (2019), 81-109.
doi: 10.1007/s10711-018-0384-8. |
| [35] |
S. Labbé,
Markov partitions for toral $\mathbb{Z}^2$-rotations featuring Jeandel–Rao Wang shift and model sets, Annales Henri Lebesgue, 4 (2021), 283-324.
doi: 10.5802/ahl.73. |
| [36] |
S. Labbé,
Substitutive structure of Jeandel-Rao aperiodic tilings, Discrete Comput. Geom., 65 (2021), 800-855.
doi: 10.1007/s00454-019-00153-3. |
| [37] |
S. Labbé, Optional SageMath Package $\texttt{slabbe}$ (Version 0.6.2), https://pypi.python.org/pypi/slabbe/, 2020. |
| [38] |
D. Lind, Multi-dimensional symbolic dynamics, in Symbolic Dynamics and its Applications, Proc. Sympos. Appl. Math., 60, Amer. Math. Soc., Providence, RI, 2004, 61–79.
doi: 10.1090/psapm/060/2078846. |
| [39] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511626302.
|
| [40] |
M. Lothaire, Combinatorics on Words, Encyclopedia of Mathematics and its Applications,
Cambridge University Press, Cambridge, 1997. |
| [41] |
M. Morse and G. A. Hedlund,
Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42.
doi: 10.2307/2371431. |
| [42] |
M. Queffélec, Substitution dynamical systems—spectral analysis, 2$^{nd}$ edition, Lecture Notes in Mathematics, 1294, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-11212-6. |
| [43] |
G. Rauzy, Une généralisation du développement en fraction continue, in Séminaire Delange-Pisot-Poitou, 18e année: 1976/77, Théorie des Nombres, Fasc. 1, Secrétariat Math., Paris, 1977, 16 pp. |
| [44] |
G. Rauzy,
Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.
doi: 10.4064/aa-34-4-315-328. |
| [45] |
G. Rauzy,
Nombres algébriques et substitutions, Bull. Soc. Math. France, 110 (1982), 147-178.
doi: 10.24033/bsmf.1957. |
| [46] |
Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.2), http://www.sagemath.org, 2020. |
| [47] |
K. Schmidt, Multi–dimensional symbolic dynamical systems, in Codes, Systems, and Graphical Models (Minneapolis, MN, 1999), IMA Vol. Math. Appl., 123, Springer, New York, 2001, 67–82.
doi: 10.1007/978-1-4613-0165-3_3. |
| [48] |
R. E. Schwartz, Outer billiards, quarter turn compositions, and polytope exchange transformations, (2011). Available from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1778&rep=rep1&type=pdf. |
| [49] |
R. E. Schwartz, The Octogonal PETs, Mathematical Surveys and Monographs, 197, American Mathematical Society, Providence, RI, 2014.
doi: 10.1090/surv/197. |
| [50] |
J. G. Sinaĭ,
Markov partitions and Y-diffeomorphisms, Funkcional. Anal. i Priložen, 2 (1968), 64-89.
|
| [51] |
J. M. Thuswaldner, $S$-adic sequences: A bridge between dynamics, arithmetic, and geometry, in Substitution and Tiling Dynamics: Introduction to Self-Inducing Structures, Lecture notes in Math., 2273, Springer, Cham, 2020, 97–191.
doi: 10.1007/978-3-030-57666-0_3. |
| [52] |
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. |
| [53] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. |
| [54] |
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. |
show all references
References:
| [1] |
R. L. Adler,
Symbolic dynamics and Markov partitions, Bull. Amer. Math. Soc. (N.S.), 35 (1998), 1-56.
doi: 10.1090/S0273-0979-98-00737-X. |
| [2] |
R. L. Adler and B. Weiss, Similarity of Automorphisms of the Torus, Memoirs of the American Mathematical Society, 98, American Mathematical Society, Providence, R.I., 1970. |
| [3] |
S. Akiyama, M. Barge, V. Berthé, J.-Y. Lee and A. Siegel, On the Pisot substitution conjecture, in Mathematics of Aperiodic Order, Progr. Math., 309, Birkhäuser/Springer, Basel, 2015, 33–72.
doi: 10.1007/978-3-0348-0903-0_2. |
| [4] |
I. Alevy, R. Kenyon and R. Yi,
A family of minimal and renormalizable rectangle exchange maps, Ergodic Theory Dynam. Systems, 41 (2021), 790-817.
doi: 10.1017/etds.2019.77. |
| [5] |
P. Arnoux, Sturmian sequences, in Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Math., 1794, Springer, Berlin, 2002,143–198.
doi: 10.1007/3-540-45714-3_6. |
| [6] |
P. Arnoux, V. Berthé and S. Ito,
Discrete planes, $\Bbb Z^2$-actions, Jacobi-Perron algorithm and substitutions, Ann. Inst. Fourier (Grenoble), 52 (2002), 305-349.
doi: 10.5802/aif.1889. |
| [7] |
P. Arnoux, V. Berthé, H. Ei and S. Ito, Tilings, quasicrystals, discrete planes, generalized substitutions, and multidimensional continued fractions, in Discrete Models: Combinatorics, Computation, and Geometry (Paris, 2001), Discrete Math. Theor. Comput. Sci. Proc., AA, Maison Inform. Math. Discrèt. (MIMD), Paris, 2001, 59–78. |
| [8] |
A. Avila and G. Forni,
Weak mixing for interval exchange transformations and translation flows, Ann. of Math. (2), 165 (2007), 637-664.
doi: 10.4007/annals.2007.165.637. |
| [9] |
M. Baake, J. A. G. Roberts and R. Yassawi,
Reversing and extended symmetries of shift spaces, Discrete Contin. Dyn. Syst., 38 (2018), 835-866.
doi: 10.3934/dcds.2018036. |
| [10] |
R. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc., 66 (1966), 72 pp. |
| [11] |
V. Berthé, Arithmetic discrete planes are quasicrystals, in Discrete Geometry for Computer Imagery, Lecture Notes in Comput. Sci., 5810, Springer, Berlin, 2009, 1–12.
doi: 10.1007/978-3-642-04397-0_1. |
| [12] |
V. Berthé, J. Bourdon, T. Jolivet and A. Siegel, Generating discrete planes with substitutions, in Combinatorics on Words, Lecture Notes in Comput. Sci., 8079, Springer, Heidelberg, 2013, 58–70.
doi: 10.1007/978-3-642-40579-2_9. |
| [13] |
V. Berthé and V. Delecroix, Beyond substitutive dynamical systems: $S$-adic expansions, in Numeration and Substitution 2012, RIMS Kôkyûroku Bessatsu, B46, Res. Inst. Math. Sci. (RIMS), Kyoto, 2014, 81–123. |
| [14] |
V. Berthé, S. Ferenczi, and L. Q. Zamboni, Interactions between dynamics, arithmetics and combinatorics: The good, the bad, and the ugly, in Algebraic and Topological Dynamics, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005,333–364.
doi: 10.1090/conm/385. |
| [15] |
V. Berthé and M. Rigo, editors, Combinatorics, automata and number theory, Encyclopedia of Mathematics and its Applications, 135, Cambridge University Press, Cambridge, 2010.
doi: 10.1017/CBO9780511777653. |
| [16] |
V. Berthé, W. Steiner and J. M. Thuswaldner,
Geometry, dynamics, and arithmetic of $S$-adic shifts, Ann. Inst. Fourier (Grenoble), 69 (2019), 1347-1409.
doi: 10.5802/aif.3273. |
| [17] |
R. Bowen,
Markov partitions for Axiom A diffeomorphisms, Amer. J. Math., 92 (1970), 725-747.
doi: 10.2307/2373370. |
| [18] |
R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. |
| [19] |
R. Bowen,
Markov partitions are not smooth, Proc. Amer. Math. Soc., 71 (1978), 130-132.
doi: 10.2307/2042234. |
| [20] |
M. Boyle and J. Tomiyama,
Bounded topological orbit equivalence and $C^*$-algebras, J. Math. Soc. Japan, 50 (1998), 317-329.
doi: 10.2969/jmsj/05020317. |
| [21] |
V. Brun, Algorithmes euclidiens pour trois et quatre nombres, in Treizième Congrès des Mathématiciens Scandinaves, tenu à Helsinki 18-23 aoşt 1957, Mercators Tryckeri, Helsinki, 1958, 45–64. |
| [22] |
E. Cawley,
Smooth Markov partitions and toral automorphisms, Ergodic Theory Dynam. Systems, 11 (1991), 633-651.
doi: 10.1017/S0143385700006404. |
| [23] |
E. Charlier, T. Kärki and M. Rigo,
Multidimensional generalized automatic sequences and shape-symmetric morphic words, Discrete Math., 310 (2010), 1238-1252.
doi: 10.1016/j.disc.2009.12.002. |
| [24] |
E. M. Coven and G. A. Hedlund,
Sequences with minimal block growth, Math. Systems Theory, 7 (1973), 138-153.
doi: 10.1007/BF01762232. |
| [25] |
M. Einsiedler and K. Schmidt,
Markov partitions and homoclinic points of algebraic $\mathbf{Z}^d$-actions, Tr. Mat. Inst. Steklova, 216 (1997), 265-284.
|
| [26] |
N. P. Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, 1794, Springer-Verlag, Berlin, 2002., Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel.
doi: 10.1007/b13861. |
| [27] |
M. Hochman, Multidimensional shifts of finite type and sofic shifts, in Combinatorics, Words
and Symbolic Dynamics, Encyclopedia Math. Appl., 159, Cambridge Univ. Press, Cambridge,
2016,296–358.
doi: 10.1017/CBO9781139924733.010. |
| [28] |
M. Hochman and T. Meyerovitch,
A characterization of the entropies of multidimensional shifts of finite type, Ann. of Math. (2), 171 (2010), 2011-2038.
doi: 10.4007/annals.2010.171.2011. |
| [29] |
W. P. Hooper,
Renormalization of polygon exchange maps arising from corner percolation, Invent. Math., 191 (2013), 255-320.
doi: 10.1007/s00222-012-0393-4. |
| [30] |
E. Jeandel and M. Rao, An aperiodic set of 11 Wang tiles, Adv. Comb., (2021), 37 pp.
doi: 10.19086/aic.18614. |
| [31] |
R. Kenyon and A. Vershik,
Arithmetic construction of sofic partitions of hyperbolic toral automorphisms, Ergodic Theory Dynam. Systems, 18 (1998), 357-372.
doi: 10.1017/S0143385798100445. |
| [32] |
B. Kitchens, Symbolic dynamics, group automorphisms and Markov partitions, in Real and Complex Dynamical Systems (Hillerød, 1993), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995,133–163. |
| [33] |
B. P. Kitchens, Symbolic Dynamics. One-sided, two-sided and countable state Markov shifts, Universitext, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-642-58822-8. |
| [34] |
S. Labbé,
A self-similar aperiodic set of 19 Wang tiles, Geom. Dedicata, 201 (2019), 81-109.
doi: 10.1007/s10711-018-0384-8. |
| [35] |
S. Labbé,
Markov partitions for toral $\mathbb{Z}^2$-rotations featuring Jeandel–Rao Wang shift and model sets, Annales Henri Lebesgue, 4 (2021), 283-324.
doi: 10.5802/ahl.73. |
| [36] |
S. Labbé,
Substitutive structure of Jeandel-Rao aperiodic tilings, Discrete Comput. Geom., 65 (2021), 800-855.
doi: 10.1007/s00454-019-00153-3. |
| [37] |
S. Labbé, Optional SageMath Package $\texttt{slabbe}$ (Version 0.6.2), https://pypi.python.org/pypi/slabbe/, 2020. |
| [38] |
D. Lind, Multi-dimensional symbolic dynamics, in Symbolic Dynamics and its Applications, Proc. Sympos. Appl. Math., 60, Amer. Math. Soc., Providence, RI, 2004, 61–79.
doi: 10.1090/psapm/060/2078846. |
| [39] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511626302.
|
| [40] |
M. Lothaire, Combinatorics on Words, Encyclopedia of Mathematics and its Applications,
Cambridge University Press, Cambridge, 1997. |
| [41] |
M. Morse and G. A. Hedlund,
Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42.
doi: 10.2307/2371431. |
| [42] |
M. Queffélec, Substitution dynamical systems—spectral analysis, 2$^{nd}$ edition, Lecture Notes in Mathematics, 1294, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-11212-6. |
| [43] |
G. Rauzy, Une généralisation du développement en fraction continue, in Séminaire Delange-Pisot-Poitou, 18e année: 1976/77, Théorie des Nombres, Fasc. 1, Secrétariat Math., Paris, 1977, 16 pp. |
| [44] |
G. Rauzy,
Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.
doi: 10.4064/aa-34-4-315-328. |
| [45] |
G. Rauzy,
Nombres algébriques et substitutions, Bull. Soc. Math. France, 110 (1982), 147-178.
doi: 10.24033/bsmf.1957. |
| [46] |
Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.2), http://www.sagemath.org, 2020. |
| [47] |
K. Schmidt, Multi–dimensional symbolic dynamical systems, in Codes, Systems, and Graphical Models (Minneapolis, MN, 1999), IMA Vol. Math. Appl., 123, Springer, New York, 2001, 67–82.
doi: 10.1007/978-1-4613-0165-3_3. |
| [48] |
R. E. Schwartz, Outer billiards, quarter turn compositions, and polytope exchange transformations, (2011). Available from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1778&rep=rep1&type=pdf. |
| [49] |
R. E. Schwartz, The Octogonal PETs, Mathematical Surveys and Monographs, 197, American Mathematical Society, Providence, RI, 2014.
doi: 10.1090/surv/197. |
| [50] |
J. G. Sinaĭ,
Markov partitions and Y-diffeomorphisms, Funkcional. Anal. i Priložen, 2 (1968), 64-89.
|
| [51] |
J. M. Thuswaldner, $S$-adic sequences: A bridge between dynamics, arithmetic, and geometry, in Substitution and Tiling Dynamics: Introduction to Self-Inducing Structures, Lecture notes in Math., 2273, Springer, Cham, 2020, 97–191.
doi: 10.1007/978-3-030-57666-0_3. |
| [52] |
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. |
| [53] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. |
| [54] |
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. |
Figure 2.
The automorphism of $ \mathbb{R}^2/ \mathbb{Z}^2 $ defined as $ v\mapsto Mv $ admits a Markov partition
Figure 3.
For every starting point $ p\in \mathbb{R}^2 $ , the coding of the shifted lattice $ p+ \mathbb{Z}^2 $ under the polygon partition $ {\mathscr{P}}_0 $ yields a configuration which is a symbolic representation of $ p $ . We show that the set of such configurations is a shift of finite type (SFT) and hence that $ {\mathscr{P}}_0 $ is a Markov partition for the toral $ \mathbb{Z}^2 $ -rotation $ R_0 $
Figure 4.
We prove that the subshifts $ X_0\subset\Omega_0 $ and $ {\mathscr{X}}_{ {\mathscr{P}}_0, R_0} $ are equal since they have a common substitutive structure. The substitutive structure of $ X_0 $ computed in [36] and the substitutive structure of $ {\mathscr{X}}_{ {\mathscr{P}}_0, R_0} $ satisfy $ \beta_0 = \omega_0\omega_1\omega_2\omega_3 $ , $ \beta_1\beta_2 = \jmath\, \eta\, \omega_6 $ , $ \beta_3 \beta_4 \beta_5 \beta_6 \beta_7 = \omega_7 \omega_8 \omega_9 \omega_{10} \omega_{11} $ , $ \zeta = \rho $ and $ \beta_8\, \beta_9\, \tau = \rho\, \omega_ {\mathscr{U}}\, \rho^{-1} $ . We deduce that $ {\mathscr{X}}_{ {\mathscr{P}}_8, R_8} = \Omega_{12} $ , $ {\mathscr{X}}_{ {\mathscr{P}}_3, R_3} = \Omega_{7} $ , $ {\mathscr{X}}_{ {\mathscr{P}}_1, R_1} = X_{4} $ and finally $ {\mathscr{X}}_{ {\mathscr{P}}_0, R_0} = X_{0} $
Figure 5.
The polygon exchange transformation $ T $ of the rectangle $ [0, \ell_1)\times[0, \ell_2) $ as defined on the figure can be seen as a toral translation by the vector $ (\alpha_1, \alpha_2) $ on the torus $ \mathbb{R}^2/(\ell_1 \mathbb{Z}\times\ell_2 \mathbb{Z}) $
Figure 6.
The partition $ {\mathscr{P}}_0 $ of $ \mathbb{R}^2/\Gamma_0 $ into atoms associated to letters in $ {\mathscr{A}}_0 $ illustrated on a rectangular fundamental domain
Figure 7.
The set $ \{A, B, C, D\} $ forms a partition of the rectangle $ W = (0, \varphi)\times(0, 1) $ . The return time to $ W $ under the map $ R_0^{ {\boldsymbol{e}}_2} $ is 4 or 5. The orbit of $ A $ , $ B $ , $ C $ and $ D $ under $ R_0^{ {\boldsymbol{e}}_2} $ before it returns to $ W $ yields a partitions of $ \mathbb{R}^2/\Gamma_0 $ . The first return map $ \widehat{R_0^{ {\boldsymbol{e}}_2}}|_W $ is equivalent to a toral translation by the vector $ (\frac{1}{\varphi}, \frac{1}{\varphi^2}) $ on $ \mathbb{R}^2/\Gamma_1 $
Figure 8.
The return time to $ W $ under map $ R_0^{ {\boldsymbol{e}}_1} $ is always 1. The first return map $ \widehat{R_0^{ {\boldsymbol{e}}_1}}|_W $ is equivalent to a toral translation by the vector $ (1, 0) $ on $ \mathbb{R}^2/\Gamma_1 $
Figure 9.
The partition $ {\mathscr{P}}_1: = \widehat{ {\mathscr{P}}_0}|_W $ of $ \mathbb{R}^2/\Gamma_1 $ into 30 convex atoms each associated to one of the 28 letters in $ {\mathscr{A}}_1 $ (indices 19 and 22 are both used twice)
Figure 10.
The partition $ {\mathscr{P}}_2 $ of $ \mathbb{R}^2/\Gamma_2 $ into atoms associated to letters in $ {\mathscr{A}}_2 $
Figure 11.
The substitutive structure of $ {\mathscr{X}}_{ {\mathscr{P}}_2, R_2} $
Figure 12.
The topological partition $ {\mathscr{P}}_ {\mathscr{U}} $ of $ \mathbb{T}^2 $ using alphabet $ \left[\!\left[ {0, {18}} \right]\!\right] $ for the atoms
| [1] |
Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002 |
| [2] |
Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801 |
| [3] |
Rostislav Grigorchuk, Volodymyr Nekrashevych. Self-similar groups, operator algebras and Schur complement. Journal of Modern Dynamics, 2007, 1 (3) : 323-370. doi: 10.3934/jmd.2007.1.323 |
| [4] |
Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198 |
| [5] |
Anna Chiara Lai, Paola Loreti. Self-similar control systems and applications to zygodactyl bird's foot. Networks and Heterogeneous Media, 2015, 10 (2) : 401-419. doi: 10.3934/nhm.2015.10.401 |
| [6] |
Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036 |
| [7] |
D. G. Aronson. Self-similar focusing in porous media: An explicit calculation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1685-1691. doi: 10.3934/dcdsb.2012.17.1685 |
| [8] |
G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically self-similar dynamics. Conference Publications, 2005, 2005 (Special) : 131-141. doi: 10.3934/proc.2005.2005.131 |
| [9] |
Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101 |
| [10] |
Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks and Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857 |
| [11] |
F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91 |
| [12] |
Shota Sato, Eiji Yanagida. Singular backward self-similar solutions of a semilinear parabolic equation. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 897-906. doi: 10.3934/dcdss.2011.4.897 |
| [13] |
Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 |
| [14] |
Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313 |
| [15] |
L. Olsen. Rates of convergence towards the boundary of a self-similar set. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 799-811. doi: 10.3934/dcds.2007.19.799 |
| [16] |
Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703 |
| [17] |
Hyungjin Huh. Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations and Control Theory, 2018, 7 (1) : 53-60. doi: 10.3934/eect.2018003 |
| [18] |
Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637 |
| [19] |
K. T. Joseph, Philippe G. LeFloch. Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits. Communications on Pure and Applied Analysis, 2002, 1 (1) : 51-76. doi: 10.3934/cpaa.2002.1.51 |
| [20] |
Meiyue Jiang, Juncheng Wei. $2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 785-803. doi: 10.3934/dcds.2016.36.785 |
2021 Impact Factor: 0.641
Tools
Metrics
Other articles
by authors
[Back to Top]