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.  Google Scholar

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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

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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

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

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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

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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

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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

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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

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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

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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 $\lambda + \delta > 1$
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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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