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  2021, 17: 33-63. doi: 10.3934/jmd.2021002

Dynamics of 2-interval piecewise affine maps and Hecke-Mahler series

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  July 20, 2019 Revised  July 30, 2020

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.

Keywords: Piecewise affine maps, rotation numbers, Sturmian sequences, Hecke-Mahler series.
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
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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

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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.  Google Scholar

[6]

J. Brémont, Dynamics of injective quasi-contractions, Ergodic. Theory Dynam. Systems, 26 (2006), 19-44.  doi: 10.1017/S0143385705000386.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[9]

R. Coutinho, Dinâmica simbólica linear, Ph.D Thesis, Instituto Superior Técnico, Universidade Técnica de Lisboa, 1999. Google Scholar

[10]

L. V. Danilov, Certain classes of transcendental numbers, Math. Zametki, 12 (1972), 149-154.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. Hata, Neurons–A Mathematical Ignition, Series on Number Theory and its Applications, 9, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. Laurent and A. Nogueira, Rotation number of contracted rotations, J. Mod. Dyn., 12 (2018), 175-191.  doi: 10.3934/jmd.2018007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. Nagumo and S. Sato, On a response characteristic of a mathematical neuron model, Kybernetik, 10 (1972), 155-164.  doi: 10.1007/BF00290514.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

M. D. Boshernitzan, Dense orbits of rationals, Proc. Amer. Math. Soc., 117 (1993), 1201-1203.  doi: 10.1090/S0002-9939-1993-1134622-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Brémont, Dynamics of injective quasi-contractions, Ergodic. Theory Dynam. Systems, 26 (2006), 19-44.  doi: 10.1017/S0143385705000386.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[9]

R. Coutinho, Dinâmica simbólica linear, Ph.D Thesis, Instituto Superior Técnico, Universidade Técnica de Lisboa, 1999. Google Scholar

[10]

L. V. Danilov, Certain classes of transcendental numbers, Math. Zametki, 12 (1972), 149-154.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. Hata, Neurons–A Mathematical Ignition, Series on Number Theory and its Applications, 9, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. Laurent and A. Nogueira, Rotation number of contracted rotations, J. Mod. Dyn., 12 (2018), 175-191.  doi: 10.3934/jmd.2018007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. Nagumo and S. Sato, On a response characteristic of a mathematical neuron model, Kybernetik, 10 (1972), 155-164.  doi: 10.1007/BF00290514.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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