Continued Fractions and Dynamics

Abstract

Several links between continued fractions and classical and less classical constructions in dynamical systems theory are presented and discussed.


Share and Cite:

Isola, S. (2014) Continued Fractions and Dynamics. Applied Mathematics, 5, 1067-1090. doi: 10.4236/am.2014.57101.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Khinchin, A.Y. (1964) Continued Fractions. The University of Chicago Press, Chicago.
[2] Hardy, G. H. and Wright, E. M. (1979) An Introduction to the Theory of Numbers. Oxford University Press, Oxford.
[3] Hensley, D. (2006) Continued Fractions. World Scientific, Singapore City.
[4] Adamczewski, B. and Allouche, J.P. (2007) Reversal and Palindromes in Continued Fractions. Theoretical Computer Science, 380, 220-237.
http://dx.doi.org/10.1016/j.tcs.2007.03.017
[5] Farey, J. (1816) On a Curious Property of Vulgar Fractions. London and Edinburgh Philosophical Magazine and Journal of Science, 47, 385-386.
[6] Appelgate, H. and Onishi, H. (1983) The Slow Continued Fraction Algorithm via 2 × 2 Integer Matrices. The American Mathematical Monthly, 90, 443-455.
http://dx.doi.org/10.2307/2975721
[7] Lagarias, J.C. (2001) The Farey Shift and the Minkowski -Function. Preprint.
[8] Apostol, T. (1976) Modular Functions and Dirichlet Series in Number Theory. Graduate Texts in Mathematic 41, Springer, New York.
http://dx.doi.org/10.1007/978-1-4684-9910-0
[9] Bonanno, C. and Isola, S. (2009) A Renormalization Approach to Irrational Rotations. Annali di Matematica Pura ed Applicata, 188, 247-267.
http://dx.doi.org/10.1007/s10231-008-0074-5
[10] Alessandri, P. and Berthè, V. (1998) Three Distances Theorem and Combinatorics on Words. L’Enseignement Mathématique, 44, 103-132.
[11] Conze, J.P. and Guivarc’h, Y. (2002) Densité d’orbites d’actions de groupes linéaires et propriétés d’équidistribution de marches aléatoires. In: Burger, M. and Iozzi, A., Eds., Rigidity in Dynamics and Geometry (Cambridge 2000), Springer, Berlin, 39-76.
[12] Knauf, A. (1999) Number Theory, Dynamical Systems and Statistical Mechanics. Reviews in Mathematical Physics, 11, 1027-1060.
http://dx.doi.org/10.1142/S0129055X99000325
[13] Salem, R. (1943) On Some Singular Monotone Functions Which Are Strictly Increasing. Transactions of the American Mathematical Society, 53, 427-439.
http://dx.doi.org/10.1090/S0002-9947-1943-0007929-6
[14] Kinney, J.R. (1960) A Note on a Singular Function of Minkowski. Proceedings of the American Mathematical Society, 11, 788-794.
[15] Bonanno, C. and Isola, S. (2009) Orderings of the Rationals and Dynamical Systems. Colloquium Mathematicum, 116, 165-189.
http://dx.doi.org/10.4064/cm116-2-3
[16] Alkauskas, G. (2010) The Minkowski Question Mark Function: Explicit Series for the Dyadic Period Function and Moments. Mathematics and Computation, 79, 383-418.
http://dx.doi.org/10.1090/S0025-5718-09-02263-7
[17] Viader, P., Paradis, J. and Bibiloni, L. (2001) A New Light on Minkowski’s (x) function. Journal of Number Theory, 73, 212-227.
http://dx.doi.org/10.1006/jnth.1998.2294
[18] Gouezel, S. (2009) Local Limit Theorem for Non-Uniformly Partially Hyperbolic Skew-Products and Farey Sequences. Duke Mathematical Journal, 147, 192-284.
http://dx.doi.org/10.1215/00127094-2009-011
[19] Bonanno, C., Carminati, C., Isola, S. and Tiozzo, G. (2013) Dynamics of Continued Fractions and Kneading Sequences of Unimodal Maps. Discrete and Continuous Dynamical Systems, 33, 1313-1332.
http://dx.doi.org/10.3934/dcds.2013.33.1313
[20] Lewis, J.B. and Zagier, D. (1997) Period Functions and the Selberg Zeta Function for the Modular Group. In: Drouffe, D.M. and Zuber, J.B., Eds., The Mathematical Beauty of Physics, Advanced Series in Mathematical Physics, Vol. 24, World Scientific, River Edge, 83-97.
[21] Isola, S. (2002) On the Spectrum of Farey and Gauss Maps. Nonlinearity, 15, 1521-1539.
http://dx.doi.org/10.1088/0951-7715/15/5/310
[22] Lewis, J.B. (1997) Spaces of Holomorphic Functions Equivalent to Even Maass Cusp forms. Inventiones Mathematicae, 127, 271-306.
http://dx.doi.org/10.1007/s002220050120
[23] Bonanno, C. and Isola, S. (2014) A Thermodynamic Approach to Two-Variable Ruelle and Selberg Zeta Functions via the Farey Map. Nonlinearity, to appear.
[24] Bowen, R. (1976) Equilibrium States and Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, 470.
[25] Ruelle, D. (1978) Thermodynamic Formalism. Addison-Wesley Publishing Company, Boston.
[26] Isola, S. (2003) On Systems with Finite Ergodic Degree. Far East Journal of Dynamical Systems, 5, 1-62.
[27] Knauf, A. (1998) The Number-Theoretical Spin Chain and the Riemann Zeros. Communications in Mathematical Physics, 196, 703-731.
http://dx.doi.org/10.1007/s002200050441
[28] Knauf, A. (1993) On a Ferromagnetic Spin Chain. Communications in Mathematical Physics, 153, 77-115.
http://dx.doi.org/10.1007/BF02099041
[29] Kotani, M. and Sunada, T. (2001) The Pressure and Higher Correlations for an ANOSOV Diffeomorphism. Ergodic Theory and Dynamical Systems, 21, 807-821.
http://dx.doi.org/10.1017/S0143385701001407
[30] Thaler, M. (1995) A Limit Theorem for the Perron-Frobenius Operator of Transformations on [0,1] with Indifferent Fixed Points. Israel Journal of Mathematics, 91, 111-127. http://dx.doi.org/10.1007/BF02761642
[31] Degli Esposti, M., Isola, S. and Knauf, A. (2007) Generalized Farey Trees, Transfer Operators and Phase Transitions. Communications in Mathematical Physics, 275, 297-329.
http://dx.doi.org/10.1007/s00220-007-0294-3
[32] Boca, F. (2007) Products of Matrices and and the Distribution of Reduced Quadratic Irrationals. Journal für die reine und angewandte Mathematik, 2007, 149-165.
[33] Bonanno, C., Graffi, S. and Isola, S. (2008) Spectral Analysis of Transfer Operators Associated to Farey Fractions. Rendiconti Lincei-Matematica e Applicazioni, 19, 1-23.
http://dx.doi.org/10.4171/RLM/505
[34] Contucci, P. and Knauf, A. (1997) The Phase Transition of the Number-Theoretical Spin Chain. Forum Mathematicum, 9, 547-567.
http://dx.doi.org/10.1515/form.1997.9.547

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.