AM  Vol.5 No.7 , April 2014
Continued Fractions and Dynamics

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

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

[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.

[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.

[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.

[9]   Bonanno, C. and Isola, S. (2009) A Renormalization Approach to Irrational Rotations. Annali di Matematica Pura ed Applicata, 188, 247-267.

[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.

[13]   Salem, R. (1943) On Some Singular Monotone Functions Which Are Strictly Increasing. Transactions of the American Mathematical Society, 53, 427-439.

[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.

[16]   Alkauskas, G. (2010) The Minkowski Question Mark Function: Explicit Series for the Dyadic Period Function and Moments. Mathematics and Computation, 79, 383-418.

[17]   Viader, P., Paradis, J. and Bibiloni, L. (2001) A New Light on Minkowski’s (x) function. Journal of Number Theory, 73, 212-227.

[18]   Gouezel, S. (2009) Local Limit Theorem for Non-Uniformly Partially Hyperbolic Skew-Products and Farey Sequences. Duke Mathematical Journal, 147, 192-284.

[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.

[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.

[22]   Lewis, J.B. (1997) Spaces of Holomorphic Functions Equivalent to Even Maass Cusp forms. Inventiones Mathematicae, 127, 271-306.

[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.

[28]   Knauf, A. (1993) On a Ferromagnetic Spin Chain. Communications in Mathematical Physics, 153, 77-115.

[29]   Kotani, M. and Sunada, T. (2001) The Pressure and Higher Correlations for an ANOSOV Diffeomorphism. Ergodic Theory and Dynamical Systems, 21, 807-821.

[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.

[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.

[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.

[34]   Contucci, P. and Knauf, A. (1997) The Phase Transition of the Number-Theoretical Spin Chain. Forum Mathematicum, 9, 547-567.