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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 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.

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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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

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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