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  2020, 16: 109-153. doi: 10.3934/jmd.2020005

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

Joshua P. Bowman 1, and  Slade Sanderson 2,

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.

Keywords: Surface, homothety, geodesic, lamination, Veech group, affine interval exchange, cutting sequence, Sturmian sequence, continued fraction, Hausdorff dimension.
Mathematics Subject Classification: Primary: 37E35; Secondary: 11A55, 37E05, 37B10.
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.

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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.

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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. Coutinho, B. Fernandez, R. 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. Duryev, C. 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. Marmi, P. 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. Coutinho, B. Fernandez, R. 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. Duryev, C. 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. Marmi, P. 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 $
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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
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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) $
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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 $
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Figure 5.  The image of $ \Upsilon_{s,\xi}^+ $ for $ 0<\xi<1 $
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Figure 6.  Stacking diagram for the word $ BAABA $
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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
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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
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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
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