American Institute of Mathematical Sciences

Advanced
  2018, 12: 175-191. doi: 10.3934/jmd.2018007

Rotation number of contracted rotations

Michel Laurent and  Arnaldo Nogueira

Aix-Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, 163 avenue de Luminy, Case 907, 13288, Marseille Cédex 9, France

Received  April 18, 2017 Revised  February 28, 2018 Published  June 2018

Let $0<\lambda<1$. We consider the one-parameter family of circle $\lambda$-affine contractions $f_\delta:x \in [0,1) \mapsto \lambda x + \delta \; {\rm mod}\,1 $, where $0 \le \delta <1$. Let $\rho$ be the rotation number of the map $f_\delta$. We will give some numerical relations between the values of $\lambda,\delta$ and $\rho$, essentially using Hecke-Mahler series and a tree structure. When both parameters $\lambda$ and $\delta$ are algebraic numbers, we show that $\rho$ is a rational number. Moreover, in the case $\lambda$ and $\delta$ are rational, we give an explicit upper bound for the height of $\rho$ under some assumptions on $\lambda$.

Keywords: Piecewise contraction, rotation number, Hecke-Mahler series.
Mathematics Subject Classification: Primary: 11J91, 37E05; Secondary: 37E45.
Citation: Michel Laurent, Arnaldo Nogueira. Rotation number of contracted rotations. Journal of Modern Dynamics, 2018, 12: 175-191. doi: 10.3934/jmd.2018007
References:
[1]

B. Adamcewski and Y. Bugeaud, Nombres réels de complexité sous-linéaire: Mesures d'irrationalité et de transcendance, J. Reine Angew. Math., 658 (2011), 65-98.  doi: 10.1515/CRELLE.2011.061.  Google Scholar

[2]

A. S. Besicovitch and S. J. Taylor, On the complementary intervals of a linear closed set of zero Lebesgue measure, Journal of the London Math. Soc., 29 (1954), 449-459.  doi: 10.1112/jlms/s1-29.4.449.  Google Scholar

[3]

P. E. Böhmer, Über die Transzendenz gewisser dyadischer Brüche, Math. Ann., 96 (1927), 367-377.  doi: 10.1007/BF01209172.  Google Scholar

[4]

J. Brémont, Dynamics of injective quasi-contractions, Ergodic. Th. and Dyn. Syst., 26 (2006), 19-44.  doi: 10.1017/S0143385705000386.  Google Scholar

[5]

Y. Bugeaud, Dynamique de certaines applications contractantes. linéaires par morceaux, sur [0,1], C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 575-578.   Google Scholar

[6]

Y. Bugeaud and J.-P. Conze, Calcul de la dynamique d'une classe de transformations linéaires contractantes mod 1 et arbre de Farey, Acta Arithmetica, 88 (1999), 201-218.  doi: 10.4064/aa-88-3-201-218.  Google Scholar

[7]

R. Coutinho, Dinâmica Simbólica Linear, Ph. D. Thesis, Instituto Superior Técnico, Universidade Técnica de Lisboa, February 1999. Google Scholar

[8]

E. J. Ding and P. C. Hemmer, Exact treatment of mode locking for a piecewise linear map, J. Statist. Phys., 46 (1987), 99-110.  doi: 10.1007/BF01010333.  Google Scholar

[9]

O. Feely and L. O. Chua, The effect of integrator leak in Σ-Δ modulation, IEEE Transactions on Circuits and Systems, 38 (1991), 1293-1305.  doi: 10.1109/31.99158.  Google Scholar

[10]

J.-M. Gambaudo and C. Tresser, On the dynamics of quasi-contractions, Bol. Coc. Bras. Mat., 19 (1988), 61-114.  doi: 10.1007/BF02584821.  Google Scholar

[11]

A. Lasota and M. C. Mackey, Noise and statistical periodicity, Physica, 28D (1987), 143-154.  doi: 10.1016/0167-2789(87)90125-4.  Google Scholar

[12]

J. H. Loxton and A. J. van der Poorten, Arithmetic properties of certain functions in several variables Ⅲ, Bull. Austral. Math. Soc., 16 (1977), 15-47.  doi: 10.1017/S0004972700022978.  Google Scholar

[13]

J. H. Loxton and A. J. van der Poorten, Transcendence and algebraic independence by a method of Mahler, in Transcendence Theory: Advances and Applications (ed. A. Baker and D. W. Masser), Academic Press, 1977,211-226.  Google Scholar

[14]

J. Nagumo and S. Sato, On a response characteristic of a mathematical neuron model, Kybernetik, 10 (1972), 155-164.   Google Scholar

[15]

K. Nishioka, Mahler Functions and Transcendence, Lecture Notes in Mathematics, 1631, Springer-Verlag, Berlin, 1996. doi: 10.1007/BFb0093672.  Google Scholar

[16]

A. Nogueira and B. Pires, Dynamics of piecewise contractions of the interval, Ergodic Theory and Dynamical Systems, 35 (2015), 2198-2215.  doi: 10.1017/etds.2014.16.  Google Scholar

[17]

F. Rhodes and C. Thompson, Rotation numbers for monotone functions on the circle, J. London Math. Soc. (2), 34 (1986), 360-368.  doi: 10.1112/jlms/s2-34.2.360.  Google Scholar

show all references

References:
[1]

B. Adamcewski and Y. Bugeaud, Nombres réels de complexité sous-linéaire: Mesures d'irrationalité et de transcendance, J. Reine Angew. Math., 658 (2011), 65-98.  doi: 10.1515/CRELLE.2011.061.  Google Scholar

[2]

A. S. Besicovitch and S. J. Taylor, On the complementary intervals of a linear closed set of zero Lebesgue measure, Journal of the London Math. Soc., 29 (1954), 449-459.  doi: 10.1112/jlms/s1-29.4.449.  Google Scholar

[3]

P. E. Böhmer, Über die Transzendenz gewisser dyadischer Brüche, Math. Ann., 96 (1927), 367-377.  doi: 10.1007/BF01209172.  Google Scholar

[4]

J. Brémont, Dynamics of injective quasi-contractions, Ergodic. Th. and Dyn. Syst., 26 (2006), 19-44.  doi: 10.1017/S0143385705000386.  Google Scholar

[5]

Y. Bugeaud, Dynamique de certaines applications contractantes. linéaires par morceaux, sur [0,1], C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 575-578.   Google Scholar

[6]

Y. Bugeaud and J.-P. Conze, Calcul de la dynamique d'une classe de transformations linéaires contractantes mod 1 et arbre de Farey, Acta Arithmetica, 88 (1999), 201-218.  doi: 10.4064/aa-88-3-201-218.  Google Scholar

[7]

R. Coutinho, Dinâmica Simbólica Linear, Ph. D. Thesis, Instituto Superior Técnico, Universidade Técnica de Lisboa, February 1999. Google Scholar

[8]

E. J. Ding and P. C. Hemmer, Exact treatment of mode locking for a piecewise linear map, J. Statist. Phys., 46 (1987), 99-110.  doi: 10.1007/BF01010333.  Google Scholar

[9]

O. Feely and L. O. Chua, The effect of integrator leak in Σ-Δ modulation, IEEE Transactions on Circuits and Systems, 38 (1991), 1293-1305.  doi: 10.1109/31.99158.  Google Scholar

[10]

J.-M. Gambaudo and C. Tresser, On the dynamics of quasi-contractions, Bol. Coc. Bras. Mat., 19 (1988), 61-114.  doi: 10.1007/BF02584821.  Google Scholar

[11]

A. Lasota and M. C. Mackey, Noise and statistical periodicity, Physica, 28D (1987), 143-154.  doi: 10.1016/0167-2789(87)90125-4.  Google Scholar

[12]

J. H. Loxton and A. J. van der Poorten, Arithmetic properties of certain functions in several variables Ⅲ, Bull. Austral. Math. Soc., 16 (1977), 15-47.  doi: 10.1017/S0004972700022978.  Google Scholar

[13]

J. H. Loxton and A. J. van der Poorten, Transcendence and algebraic independence by a method of Mahler, in Transcendence Theory: Advances and Applications (ed. A. Baker and D. W. Masser), Academic Press, 1977,211-226.  Google Scholar

[14]

J. Nagumo and S. Sato, On a response characteristic of a mathematical neuron model, Kybernetik, 10 (1972), 155-164.   Google Scholar

[15]

K. Nishioka, Mahler Functions and Transcendence, Lecture Notes in Mathematics, 1631, Springer-Verlag, Berlin, 1996. doi: 10.1007/BFb0093672.  Google Scholar

[16]

A. Nogueira and B. Pires, Dynamics of piecewise contractions of the interval, Ergodic Theory and Dynamical Systems, 35 (2015), 2198-2215.  doi: 10.1017/etds.2014.16.  Google Scholar

[17]

F. Rhodes and C. Thompson, Rotation numbers for monotone functions on the circle, J. London Math. Soc. (2), 34 (1986), 360-368.  doi: 10.1112/jlms/s2-34.2.360.  Google Scholar

Figure 1.  A plot of $f_{\lambda, \delta}: I \to I$ , where $\lambda + \delta > 1$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Plot of $F_{1/2, 3/4}(x) $ in the interval $-1\le x < 1$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Arek Goetz. Dynamics of a piecewise rotation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 593-608. doi: 10.3934/dcds.1998.4.593
[2]

Wenxian Shen. Global attractor and rotation number of a class of nonlinear noisy oscillators. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 597-611. doi: 10.3934/dcds.2007.18.597
[3]

Song-Mei Huan, Xiao-Song Yang. On the number of limit cycles in general planar piecewise linear systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2147-2164. doi: 10.3934/dcds.2012.32.2147
[4]

Qiudong Wang. The diffusion time of the connecting orbit around rotation number zero for the monotone twist maps. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 255-274. doi: 10.3934/dcds.2000.6.255
[5]

Tiago de Carvalho, Bruno Freitas. Birth of an arbitrary number of T-singularities in 3D piecewise smooth vector fields. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4851-4861. doi: 10.3934/dcdsb.2019034
[6]

Dieter Mayer, Tobias Mühlenbruch, Fredrik Strömberg. The transfer operator for the Hecke triangle groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2453-2484. doi: 10.3934/dcds.2012.32.2453
[7]

Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581
[8]

Frédéric Naud, Anke Pohl, Louis Soares. Fractal Weyl bounds and Hecke triangle groups. Electronic Research Announcements, 2019, 26: 24-35. doi: 10.3934/era.2019.26.003
[9]

Peter Giesl, Holger Wendland. Construction of a contraction metric by meshless collocation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3843-3863. doi: 10.3934/dcdsb.2018333
[10]

Armengol Gasull, Francesc Mañosas. Subseries and signed series. Communications on Pure & Applied Analysis, 2019, 18 (1) : 479-492. doi: 10.3934/cpaa.2019024
[11]

Peter Giesl. Converse theorem on a global contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5339-5363. doi: 10.3934/dcds.2019218
[12]

Danny Calegari, Alden Walker. Ziggurats and rotation numbers. Journal of Modern Dynamics, 2011, 5 (4) : 711-746. doi: 10.3934/jmd.2011.5.711
[13]

Xavier Buff, Nataliya Goncharuk. Complex rotation numbers. Journal of Modern Dynamics, 2015, 9: 169-190. doi: 10.3934/jmd.2015.9.169
[14]

Sangtae Jeong, Chunlan Li. Measure-preservation criteria for a certain class of 1-lipschitz functions on Zp in mahler's expansion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3787-3804. doi: 10.3934/dcds.2017160
[15]

Christopher Cleveland. Rotation sets for unimodal maps of the interval. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 617-632. doi: 10.3934/dcds.2003.9.617
[16]

David Cowan. A billiard model for a gas of particles with rotation. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 101-109. doi: 10.3934/dcds.2008.22.101
[17]

Mads Kyed. On a mapping property of the Oseen operator with rotation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1315-1322. doi: 10.3934/dcdss.2013.6.1315
[18]

Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056
[19]

T. L. van Noorden, I. S. Pop, M. Röger. Crystal dissolution and precipitation in porous media: L$^1$-contraction and uniqueness. Conference Publications, 2007, 2007 (Special) : 1013-1020. doi: 10.3934/proc.2007.2007.1013
[20]

Dirk Aeyels, Filip De Smet, Bavo Langerock. Area contraction in the presence of first integrals and almost global convergence. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 135-157. doi: 10.3934/dcds.2007.18.135

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (126)
  • HTML views (1082)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences