American Institute of Mathematical Sciences

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  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.

[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.

[3]

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

[4]

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

[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. 

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[15]

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

[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.

[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.

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.

[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.

[3]

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

[4]

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

[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. 

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[15]

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

[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.

[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.

Figure 1.  A plot of $f_{\lambda, \delta}: I \to I$ , where $\lambda + \delta > 1$
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Figure 2.  Plot of $F_{1/2, 3/4}(x) $ in the interval $-1\le x < 1$
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