Figure 1.
A plot of $ f_{\lambda, \mu, \delta} $
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On mixing and sparse ergodic theorems
2021, 17: 33-63.
doi: 10.3934/jmd.2021002
Dynamics of 2-interval piecewise affine maps and Hecke-Mahler series
Aix-Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, 163 Avenue de Luminy, Case 907, 13288, Marseille Cédex 9, France |
Let $ f : [0,1)\rightarrow [0,1) $ be a $ 2 $-interval piecewise affine increasing map which is injective but not surjective. Such a map $ f $ has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of $ f $ thanks to two specific functions $ {\boldsymbol{\delta}} $ and $ \phi $ depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of $ f $ is rational, whenever the three parameters are all algebraic numbers, extending thus the main result of [
Mathematics Subject Classification:
Primary: 11J91, 37E05; Secondary: 37E45.
Citation:
Michel Laurent, Arnaldo Nogueira. Dynamics of 2-interval piecewise affine maps and Hecke-Mahler series. Journal of Modern Dynamics,
2021, 17: 33-63.
doi: 10.3934/jmd.2021002
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References:
| [1] |
W. W. Adams and J. L. Davison,
A remarkable class of continued fractions, Proc. Amer. Math. Soc., 65 (1977), 194-198.
doi: 10.1090/S0002-9939-1977-0441879-4. |
| [2] |
P. E. Böhmer,
$\ddot{U}ber$ die Transzendenz gewisser dyadischer Br$\ddot{u}$che, Math. Ann., 96 (1927), 367-377.
doi: 10.1007/BF01209172. |
| [3] |
M. D. Boshernitzan,
Dense orbits of rationals, Proc. Amer. Math. Soc., 117 (1993), 1201-1203.
doi: 10.1090/S0002-9939-1993-1134622-6. |
| [4] |
J. P. Bowman and S. Sanderson,
Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces, J. Mod. Dyn., 16 (2020), 109-153.
doi: 10.3934/jmd.2020005. |
| [5] |
J. M. Borwein and P. B. Borwein,
On the generating function of the integer part: $[ n \alpha + \gamma]$, J. Number Theory, 43 (1993), 293-318.
doi: 10.1006/jnth.1993.1023. |
| [6] |
J. Brémont,
Dynamics of injective quasi-contractions, Ergodic. Theory Dynam. Systems, 26 (2006), 19-44.
doi: 10.1017/S0143385705000386. |
| [7] |
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.
|
| [8] |
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. |
| [9] |
R. Coutinho, Dinâmica simbólica linear, Ph.D Thesis, Instituto Superior Técnico, Universidade Técnica de Lisboa, 1999. |
| [10] |
L. V. Danilov,
Certain classes of transcendental numbers, Math. Zametki, 12 (1972), 149-154.
|
| [11] |
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. |
| [12] |
O. Feely and L. O. Chua,
The effect of integrator leak in $\Sigma-\Delta$ modulation, IEEE Transactions on Circuits and Systems, 38 (1991), 1293-1305.
doi: 10.1109/31.99158. |
| [13] |
M. Hata, Neurons–A Mathematical Ignition, Series on Number Theory and its Applications, 9, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. |
| [14] |
S. Janson and A. Öberg,
A piecewise contractive dynamical system and Phragmén's election method, Bull. Soc. Math. France, 147 (2019), 395-441.
doi: 10.24033/bsmf.2787. |
| [15] |
T. Komatsu,
A certain power series and the inhomogeneous continued fraction expansions, J. Number Theory, 59 (1996), 291-312.
doi: 10.1006/jnth.1996.0099. |
| [16] |
M. Laurent and A. Nogueira,
Rotation number of contracted rotations, J. Mod. Dyn., 12 (2018), 175-191.
doi: 10.3934/jmd.2018007. |
| [17] |
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. |
| [18] |
J. H. Loxton and A. J. van der Poorten, Transcendence and algebraic independence by a method of Mahler, in Transcendence Theory: Advances and Applications (Proc. Conf., Univ. Cambridge, Cambridge, 1976), Academic Press, London, 1977,211–226. |
| [19] |
J. Nagumo and S. Sato,
On a response characteristic of a mathematical neuron model, Kybernetik, 10 (1972), 155-164.
doi: 10.1007/BF00290514. |
| [20] |
K. Nishioka, Mahler Functions and Transcendence, Springer Lecture Notes in Mathematics, 1631, Springer-Verlag, Berlin, 1996.
doi: 10.1007/BFb0093672. |
| [21] |
K. Nishioka, I. Shiokawa and J. Tamura,
Arithmetical properties of a certain power series, J. Number Theory, 42 (1992), 61-87.
doi: 10.1016/0022-314X(92)90109-3. |
| [22] |
A. Nogueira and B. Pires,
Dynamics of piecewise contractions of the interval, Ergodic Theory Dynam. Systems, 35 (2015), 2198-2215.
doi: 10.1017/etds.2014.16. |
| [23] |
A. Nogueira, B. Pires and R. A. Rosales,
Topological dynamics of piecewise $\lambda$-affine maps, Ergodic Theory Dynam. Systems, 38 (2018), 1876-1893.
doi: 10.1017/etds.2016.104. |
| [24] |
F. Rhodes and C. L. 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. |
| [25] |
F. Rhodes and C. L. Thompson,
Topologies and rotation numbers for families of monotone functions on the circle, J. London Math. Soc. (2), 43 (1991), 156-170.
doi: 10.1112/jlms/s2-43.1.156. |
show all references
References:
| [1] |
W. W. Adams and J. L. Davison,
A remarkable class of continued fractions, Proc. Amer. Math. Soc., 65 (1977), 194-198.
doi: 10.1090/S0002-9939-1977-0441879-4. |
| [2] |
P. E. Böhmer,
$\ddot{U}ber$ die Transzendenz gewisser dyadischer Br$\ddot{u}$che, Math. Ann., 96 (1927), 367-377.
doi: 10.1007/BF01209172. |
| [3] |
M. D. Boshernitzan,
Dense orbits of rationals, Proc. Amer. Math. Soc., 117 (1993), 1201-1203.
doi: 10.1090/S0002-9939-1993-1134622-6. |
| [4] |
J. P. Bowman and S. Sanderson,
Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces, J. Mod. Dyn., 16 (2020), 109-153.
doi: 10.3934/jmd.2020005. |
| [5] |
J. M. Borwein and P. B. Borwein,
On the generating function of the integer part: $[ n \alpha + \gamma]$, J. Number Theory, 43 (1993), 293-318.
doi: 10.1006/jnth.1993.1023. |
| [6] |
J. Brémont,
Dynamics of injective quasi-contractions, Ergodic. Theory Dynam. Systems, 26 (2006), 19-44.
doi: 10.1017/S0143385705000386. |
| [7] |
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.
|
| [8] |
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. |
| [9] |
R. Coutinho, Dinâmica simbólica linear, Ph.D Thesis, Instituto Superior Técnico, Universidade Técnica de Lisboa, 1999. |
| [10] |
L. V. Danilov,
Certain classes of transcendental numbers, Math. Zametki, 12 (1972), 149-154.
|
| [11] |
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. |
| [12] |
O. Feely and L. O. Chua,
The effect of integrator leak in $\Sigma-\Delta$ modulation, IEEE Transactions on Circuits and Systems, 38 (1991), 1293-1305.
doi: 10.1109/31.99158. |
| [13] |
M. Hata, Neurons–A Mathematical Ignition, Series on Number Theory and its Applications, 9, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. |
| [14] |
S. Janson and A. Öberg,
A piecewise contractive dynamical system and Phragmén's election method, Bull. Soc. Math. France, 147 (2019), 395-441.
doi: 10.24033/bsmf.2787. |
| [15] |
T. Komatsu,
A certain power series and the inhomogeneous continued fraction expansions, J. Number Theory, 59 (1996), 291-312.
doi: 10.1006/jnth.1996.0099. |
| [16] |
M. Laurent and A. Nogueira,
Rotation number of contracted rotations, J. Mod. Dyn., 12 (2018), 175-191.
doi: 10.3934/jmd.2018007. |
| [17] |
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. |
| [18] |
J. H. Loxton and A. J. van der Poorten, Transcendence and algebraic independence by a method of Mahler, in Transcendence Theory: Advances and Applications (Proc. Conf., Univ. Cambridge, Cambridge, 1976), Academic Press, London, 1977,211–226. |
| [19] |
J. Nagumo and S. Sato,
On a response characteristic of a mathematical neuron model, Kybernetik, 10 (1972), 155-164.
doi: 10.1007/BF00290514. |
| [20] |
K. Nishioka, Mahler Functions and Transcendence, Springer Lecture Notes in Mathematics, 1631, Springer-Verlag, Berlin, 1996.
doi: 10.1007/BFb0093672. |
| [21] |
K. Nishioka, I. Shiokawa and J. Tamura,
Arithmetical properties of a certain power series, J. Number Theory, 42 (1992), 61-87.
doi: 10.1016/0022-314X(92)90109-3. |
| [22] |
A. Nogueira and B. Pires,
Dynamics of piecewise contractions of the interval, Ergodic Theory Dynam. Systems, 35 (2015), 2198-2215.
doi: 10.1017/etds.2014.16. |
| [23] |
A. Nogueira, B. Pires and R. A. Rosales,
Topological dynamics of piecewise $\lambda$-affine maps, Ergodic Theory Dynam. Systems, 38 (2018), 1876-1893.
doi: 10.1017/etds.2016.104. |
| [24] |
F. Rhodes and C. L. 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. |
| [25] |
F. Rhodes and C. L. Thompson,
Topologies and rotation numbers for families of monotone functions on the circle, J. London Math. Soc. (2), 43 (1991), 156-170.
doi: 10.1112/jlms/s2-43.1.156. |
Figure 2.
Plot of the map $ \rho \mapsto {\boldsymbol{\delta}}(0.9,0.8, \rho) $
Figure 4.
Plot of the function $ \phi_{0.95, 0.9, \delta, (\sqrt{5}-1)/2} $ in the range $ \;\;\;0\le y \le1 $ , where $ \delta = {\boldsymbol{\delta}}(0.95,0.9,(\sqrt{5}-1)/2) = 0.6617\dots $
Figure 7.
Dynamics of the map $ f $ with $ \zeta_0>0 $ on the left and $ \zeta_0 = 0 $ on the right. The arrows indicate the action of $ f $ on the intervals
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