\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces

Abstract Full Text(HTML) Figure(9) Related Papers Cited by
  • A homothety surface can be assembled from polygons by identifying their edges in pairs via homotheties, which are compositions of translation and scaling. We consider linear trajectories on a $ 1 $-parameter family of genus-$ 2 $ homothety surfaces. The closure of a trajectory on each of these surfaces always has Hausdorff dimension $ 1 $, and contains either a closed loop or a lamination with Cantor cross-section. Trajectories have cutting sequences that are either eventually periodic or eventually Sturmian. Although no two of these surfaces are affinely equivalent, their linear trajectories can be related directly to those on the square torus, and thence to each other, by means of explicit functions. We also briefly examine two related families of surfaces and show that the above behaviors can be mixed; for instance, the closure of a linear trajectory can contain both a closed loop and a lamination.

    Mathematics Subject Classification: Primary: 37E35; Secondary: 11A55, 37E05, 37B10.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  LEFT: The union of the rectangles $ R_s^\pm $. RIGHT: Edge identifications that produce the surface $ X_s $. Horizontal edges are identified by translations; vertical edges are identified by homotheties. $ X_s^+ $ and $ X_s^- $ are genus $ 1 $ subsurfaces with boundary. They are joined along the saddle connection $ E $

    Figure 2.  The graph of $ \Delta_s^-(x) $ when $ s = 2/3 $, drawn only for $ 0 \le x \le 1 $. The graph of $ \Delta_s^+(x) $ appears the same; the differences occur only at the jumps, which are dense but countable

    Figure 3.  "Tongues of angels:" Each tongue corresponds to a rational number in $ (0,1) $. The boundary curves are drawn using the formulas for $ \Delta_s^\pm(k/n) $

    Figure 4.  The graphs of two functions $ \Upsilon_{s,\xi}^\pm(x) $ with $ s = 0.95 $, drawn only for $ 0 \le x \le 1 $. TOP: $ \xi = (1+\sqrt5)/2 $. BOTTOM: $ \xi = \pi $

    Figure 5.  The image of $ \Upsilon_{s,\xi}^+ $ for $ 0<\xi<1 $

    Figure 6.  Stacking diagram for the word $ BAABA $

    Figure 7.  The trajectories $ \tau_\xi^- $ and $ \tau_\xi^+ $, drawn on a partial stacking diagram. LEFT: $ \xi \notin \mathbb{Q} $, as in §4.3. These trajectories are parallel and bound a strip in $ X_s $. RIGHT: $ \xi = k/n \in \mathbb{Q} $, as in §4.4. These trajectories will intersect after crossing $ k + n $ edges

    Figure 8.  LEFT: A surface $ X_s^u $ with $ 0 < u < 1 $. Lower-case letters indicate dimensions. Capital letters indicate gluings between edges. RIGHT: The surface $ X_{1/2}^{1/2} $, which was the first surface the authors considered

    Figure 9.  LEFT: The rectangles $ R_{s_1}^+ $ and $ R_{s_2}^- $. RIGHT: Edge identifications to form the surface $ X_{s_1,s_2} $. $ X_{s_1,s_2}^+ $ is isomorphic to $ X_{s_1}^+ $, and $ X_{s_1,s_2}^- $ is isomorphic to $ X_{s_2}^- $, as defined in §2.6

  • [1] D. Bailey and R. Crandall, Random generators and normal numbers, Exp. Math., 11 (2002), 527-546.  doi: 10.1080/10586458.2002.10504704.
    [2] S. Bates and A. Norton, On sets of critical values in the real line, Duke Math. J., 83 (1996), 399-413.  doi: 10.1215/S0012-7094-96-08313-1.
    [3] A. Besicovitch and S. Taylor, On the complementary intervals of a linear closed set of zero Lebesgue measure, J. London Math. Soc., 29 (1954), 449-459.  doi: 10.1112/jlms/s1-29.4.449.
    [4] A. Boulanger, C. Fougeron and S. Ghazouani, Cascades in the dynamics of affine interval exchange transformations, Erg. Th. Dyn. Sys., (2019), 1–25. doi: 10.1017/etds.2018.141.
    [5] Y. Bugeaud, Dynamique de certaines applications contractantes, linéaires par morceaux, sur [0, 1[, C.R.A.S. Série I, 317 (1993), 575-578. 
    [6] Y. Bugeaud and J.-P. Conze, Calcul de la dynamique de transformations linéaires contractantes mod 1 et arbre de Farey, Acta Arith., 88 (1999), 201-218.  doi: 10.4064/aa-88-3-201-218.
    [7] R. Coutinho, Dinâmica Simbólica Linear, Ph. D. Thesis, Instituto Superior Técnico, Universidade Técnica de Lisboa, 1999.
    [8] R. CoutinhoB. FernandezR. Lima and A. Meyroneinc, Discrete time piecewise affine models of genetic regulatory networks, J. Math. Biol., 52 (2006), 524-570.  doi: 10.1007/s00285-005-0359-x.
    [9] E. J. Ding and P. C. Hemmer, Exact treatment of mode locking for a piecewise linear map, J. Stat. Phys., 46 (1987), 99-110.  doi: 10.1007/BF01010333.
    [10] E. DuryevC. Fougeron and S. Ghazouani, Dilation surfaces and their Veech groups, J. Mod. Dyn., 14 (2019), 121-151.  doi: 10.3934/jmd.2019005.
    [11] A. Hatcher and U. Oertel, Affine lamination spaces for surfaces, Pacific J. Math., 154 (1992), 87-101.  doi: 10.2140/pjm.1992.154.87.
    [12] M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. maths. de l'I.H.É.S., 49 (1979), 5-233. 
    [13] S. Janson and A. Öberg, A piecewise contractive dynamical system and election methods, Bull. Soc. Math. de France, 147 (2019), 395-441.  doi: 10.24033/bsmf.2787.
    [14] A. Khinchin, Continued Fractions, Courier Corporation, 1964.
    [15] L. Kuipers and H. Neiderreiter, Uniform Distribution of Sequences, John Wiley & Sons, 1974.
    [16] I. Liousse, Dynamique générique des feuilletages transversalement affines des surfaces, Bull. S.M.F., 123 (1995), 493-516.  doi: 10.24033/bsmf.2268.
    [17] J. Liouville, Sur quelques séries et produits infinis, J. math. pures et appl. 2e série, 2 (1857), 433-440. 
    [18] M. Laurent and A. Nogueira, Rotation number of contracted rotations, J. Mod. Dyn., 12 (2018), 175-191.  doi: 10.3934/jmd.2018007.
    [19] S. MarmiP. Moussa and J.-C. Yoccoz, Affine interval exchange maps with a wandering interval, Proc. London Math. Soc., 100 (2010), 639-669.  doi: 10.1112/plms/pdp037.
    [20] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. A.M.S., 19 (1988), 417-431.  doi: 10.1090/S0273-0979-1988-15685-6.
    [21] W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Inv. Math., 97 (1989), 553-583.  doi: 10.1007/BF01388890.
    [22] P. Veerman, Symbolic dynamics of order-preserving orbits, Physica D: Nonlinear Phenomena, 29 (1987), 191-201.  doi: 10.1016/0167-2789(87)90055-8.
  • 加载中

Figures(9)

SHARE

Article Metrics

HTML views(1838) PDF downloads(304) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return