# American Institute of Mathematical Sciences

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. 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.
A plot of $f_{\lambda, \mu, \delta}$
Plot of the map $\rho \mapsto {\boldsymbol{\delta}}(0.9,0.8, \rho)$
A plot of $f_{\lambda, \mu, d_{\lambda,\mu}}$ for $\lambda = 1/2$, $\mu = 3$
Plot of the function $\phi_{0.95, 0.9, \delta, (\sqrt{5}-1)/2}$ in the range $\;\;\;0\le y \le1$, where $\delta = {\boldsymbol{\delta}}(0.95,0.9,(\sqrt{5}-1)/2) = 0.6617\dots$
Plot of $F_{1/2, 1/2, 3/4}(x)$ in the interval $-1\le x < 1$
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
Case $\zeta_0>0$ and Case $\zeta_0 = 0$
 [1] Jozef Bobok, Martin Soukenka. On piecewise affine interval maps with countably many laps. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 753-762. doi: 10.3934/dcds.2011.31.753 [2] Lorenzo Sella, Pieter Collins. Computation of symbolic dynamics for two-dimensional piecewise-affine maps. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 739-767. doi: 10.3934/dcdsb.2011.15.739 [3] Danny Calegari, Alden Walker. Ziggurats and rotation numbers. Journal of Modern Dynamics, 2011, 5 (4) : 711-746. doi: 10.3934/jmd.2011.5.711 [4] Xavier Buff, Nataliya Goncharuk. Complex rotation numbers. Journal of Modern Dynamics, 2015, 9: 169-190. doi: 10.3934/jmd.2015.9.169 [5] Jon Chaika, David Constantine. A quantitative shrinking target result on Sturmian sequences for rotations. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5189-5204. doi: 10.3934/dcds.2018229 [6] Joshua P. Bowman, Slade Sanderson. Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces. Journal of Modern Dynamics, 2020, 16: 109-153. doi: 10.3934/jmd.2020005 [7] Arek Goetz. Dynamics of a piecewise rotation. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 593-608. doi: 10.3934/dcds.1998.4.593 [8] Jifeng Chu, Meirong Zhang. Rotation numbers and Lyapunov stability of elliptic periodic solutions. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1071-1094. doi: 10.3934/dcds.2008.21.1071 [9] Anna Belova. Rigorous enclosures of rotation numbers by interval methods. Journal of Computational Dynamics, 2016, 3 (1) : 81-91. doi: 10.3934/jcd.2016004 [10] Changzhi Wu, Kok Lay Teo, Volker Rehbock. Optimal control of piecewise affine systems with piecewise affine state feedback. Journal of Industrial and Management Optimization, 2009, 5 (4) : 737-747. doi: 10.3934/jimo.2009.5.737 [11] Christopher Cleveland. Rotation sets for unimodal maps of the interval. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 617-632. doi: 10.3934/dcds.2003.9.617 [12] Abdelhamid Adouani, Habib Marzougui. Computation of rotation numbers for a class of PL-circle homeomorphisms. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3399-3419. doi: 10.3934/dcds.2012.32.3399 [13] Shuang Wang, Dingbian Qian. Periodic solutions of p-Laplacian equations via rotation numbers. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2117-2138. doi: 10.3934/cpaa.2021060 [14] C. Kopf. Symbol sequences and entropy for piecewise monotone transformations with discontinuities. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 299-304. doi: 10.3934/dcds.2000.6.299 [15] Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343 [16] Tiantian Wu, Xiao-Song Yang. A new class of 3-dimensional piecewise affine systems with homoclinic orbits. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5119-5129. doi: 10.3934/dcds.2016022 [17] Sigurdur F. Hafstein, Christopher M. Kellett, Huijuan Li. Computing continuous and piecewise affine lyapunov functions for nonlinear systems. Journal of Computational Dynamics, 2015, 2 (2) : 227-246. doi: 10.3934/jcd.2015004 [18] Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo. Dynamics of continued fractions and kneading sequences of unimodal maps. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1313-1332. doi: 10.3934/dcds.2013.33.1313 [19] Viviane Baladi, Sébastien Gouëzel. Banach spaces for piecewise cone-hyperbolic maps. Journal of Modern Dynamics, 2010, 4 (1) : 91-137. doi: 10.3934/jmd.2010.4.91 [20] Peter Ashwin, Xin-Chu Fu. Symbolic analysis for some planar piecewise linear maps. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1533-1548. doi: 10.3934/dcds.2003.9.1533

2021 Impact Factor: 0.641