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

Received  July 20, 2019 Revised  July 30, 2020 Published  January 2021

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 [16] dealing with the particular case of $ 2 $-interval piecewise affine contractions with constant slope.

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
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. NishiokaI. 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. NogueiraB. 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. NishiokaI. 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. NogueiraB. 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 1.  A plot of $ f_{\lambda, \mu, \delta} $
Figure 2.  Plot of the map $ \rho \mapsto {\boldsymbol{\delta}}(0.9,0.8, \rho) $
Figure 3.  A plot of $ f_{\lambda, \mu, d_{\lambda,\mu}} $ for $ \lambda = 1/2 $, $ \mu = 3 $
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 5.  Plot of $ F_{1/2, 1/2, 3/4}(x) $ in the interval $ -1\le x < 1 $
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
Figure 6.  Case $ \zeta_0>0 $ and Case $ \zeta_0 = 0 $
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