• Previous Article
    Equilibrium measures for some partially hyperbolic systems
  • JMD Home
  • This Volume
  • Next Article
    Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani's asymptotics for simple closed hyperbolic geodesics
  2020, 16: 109-153. doi: 10.3934/jmd.2020005

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

1. 

Natural Science Division, Pepperdine University, 24255 Pacific Coast Highway, Malibu, CA 90263, USA

2. 

Department of Mathematics, Kidder Hall 368, Oregon State University, Corvallis, OR 97331, USA

Received  September 15, 2018 Revised  February 13, 2020 Published  April 2020

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.

Citation: Joshua P. Bowman, Slade Sanderson. Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces. Journal of Modern Dynamics, 2020, 16: 109-153. doi: 10.3934/jmd.2020005
References:
[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.

show all references

References:
[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.

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]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457

[2]

Davit Karagulyan. Hausdorff dimension of a class of three-interval exchange maps. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1257-1281. doi: 10.3934/dcds.2020077

[3]

Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018

[4]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267

[5]

Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098

[6]

Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417

[7]

José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781

[8]

Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015

[9]

Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060

[10]

Svetlana Katok, Ilie Ugarcovici. Theory of $(a,b)$-continued fraction transformations and applications. Electronic Research Announcements, 2010, 17: 20-33. doi: 10.3934/era.2010.17.20

[11]

Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281

[12]

Svetlana Katok, Ilie Ugarcovici. Structure of attractors for $(a,b)$-continued fraction transformations. Journal of Modern Dynamics, 2010, 4 (4) : 637-691. doi: 10.3934/jmd.2010.4.637

[13]

Zhenyu Zhang, Lijia Ge, Fanxin Zeng, Guixin Xuan. Zero correlation zone sequence set with inter-group orthogonal and inter-subgroup complementary properties. Advances in Mathematics of Communications, 2015, 9 (1) : 9-21. doi: 10.3934/amc.2015.9.9

[14]

Alireza Goli, Taha Keshavarz. Just-in-time scheduling in identical parallel machine sequence-dependent group scheduling problem. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021124

[15]

Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015

[16]

Walter Briec, Bernardin Solonandrasana. Some remarks on a successive projection sequence. Journal of Industrial and Management Optimization, 2006, 2 (4) : 451-466. doi: 10.3934/jimo.2006.2.451

[17]

Ivan Dynnikov, Alexandra Skripchenko. Minimality of interval exchange transformations with restrictions. Journal of Modern Dynamics, 2017, 11: 219-248. doi: 10.3934/jmd.2017010

[18]

Kai-Uwe Schmidt, Jonathan Jedwab, Matthew G. Parker. Two binary sequence families with large merit factor. Advances in Mathematics of Communications, 2009, 3 (2) : 135-156. doi: 10.3934/amc.2009.3.135

[19]

Matthew Macauley, Henning S. Mortveit. Update sequence stability in graph dynamical systems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1533-1541. doi: 10.3934/dcdss.2011.4.1533

[20]

Wenjun Xia, Jinzhi Lei. Formulation of the protein synthesis rate with sequence information. Mathematical Biosciences & Engineering, 2018, 15 (2) : 507-522. doi: 10.3934/mbe.2018023

2020 Impact Factor: 0.848

Metrics

  • PDF downloads (261)
  • HTML views (464)
  • Cited by (0)

Other articles
by authors

[Back to Top]