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2018, 12: 175-191.
doi: 10.3934/jmd.2018007
Rotation number of contracted rotations
Aix-Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, 163 avenue de Luminy, Case 907, 13288, Marseille Cédex 9, France |
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 $\lambda$.
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
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References:
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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.
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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. |
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R. Coutinho, Dinâmica Simbólica Linear, Ph. D. Thesis, Instituto Superior Técnico, Universidade Técnica de Lisboa, February 1999. |
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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.
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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. |
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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. |
| [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. |
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J. Nagumo and S. Sato, On a response characteristic of a mathematical neuron model, Kybernetik, 10 (1972), 155-164. |
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K. Nishioka, Mahler Functions and Transcendence, Lecture Notes in Mathematics, 1631, Springer-Verlag, Berlin, 1996.
doi: 10.1007/BFb0093672. |
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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. |
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