Combinatorics and dynamical systems

The mathematical disciplines of combinatorics and dynamical systems interact in a number of ways. The ergodic theory of dynamical systems has recently been used to prove combinatorial theorems about number theory which has given rise to the field of arithmetic combinatorics. Also dynamical systems theory is heavily involved in the relatively recent field of combinatorics on words. Also combinatorial aspects of dynamical systems are studied. Dynamical systems can be defined on combinatorial objects; see for example graph dynamical system.

See also

• Symbolic dynamics
• Analytic combinatorics
• Combinatorics and physics
• Arithmetic dynamics

References

• Alsedà, Lluís; Libre, Jaume; Misiurewicz, Michał (October 2000), Combinatorial Dynamics and Entropy in Dimension One (2nd ed.), World Scientific, ISBN 978-981-02-4053-0
• Baake, Michael; Damanik, David; Putnam, Ian; Solomyak, Boris (2004), Aperiodic Order: Dynamical Systems, Combinatorics, and Operators (PDF), Banff International Research Station for Mathematical Innovation and Discovery.
• Berthé, Valérie; Ferenczi, Sébastien; Zamboni, Luca Q. (2005), "Interactions between dynamics, arithmetics and combinatorics: the good, the bad, and the ugly", Algebraic and topological dynamics, Contemp. Math., vol. 385, Providence, RI: Amer. Math. Soc., pp. 333–364, MR 2180244.
• Fauvet, F.; Mitschi, C. (2003), From combinatorics to dynamical systems: Proceedings of the Computer Algebra Conference in honor of Jean Thomann held in Strasbourg, March 22–23, 2002, IRMA Lectures in Mathematics and Theoretical Physics, vol. 3, Berlin: Walter de Gruyter & Co., ISBN 3-11-017875-3, MR 2049418.
• Fogg, N. Pytheas (2002), Fogg, N. Pytheas; Berthé, Valéré; Ferenczi, Sébastien; Mauduit, Christian; Siegel, Anne (eds.), Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, vol. 1794, Berlin: Springer-Verlag, doi:10.1007/b13861, ISBN 3-540-44141-7, MR 1970385.
• Forman, Robin (1998), "Combinatorial vector fields and dynamical systems", Mathematische Zeitschrift, 228 (4): 629–681, doi:10.1007/PL00004638, MR 1644432, S2CID 121002180.
• Kaimanovich, V.; Lodkin, A. (2006), Representation theory, dynamical systems, and asymptotic combinatorics (Papers from the conference held in St. Petersburg, June 8–13, 2004), American Mathematical Society Translations, Series 2, vol. 217, Providence, RI: American Mathematical Society, ISBN 978-0-8218-4208-9, MR 2286117.
• Latapy, Matthieu (2000), "Generalized integer partitions, tilings of zonotopes and lattices", in Krob, Daniel; Mikhalev, Alexander A. (eds.), Formal Power Series and Algebraic Combinatorics: 12th International Conference, FPSAC'00, Moscow, Russia, June 2000, Proceedings, Berlin: Springer, pp. 256–267, arXiv:math/0008022, Bibcode:2000math......8022L, MR 1798219.
• Lothaire, M. (2005), Applied combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 105, Cambridge: Cambridge University Press, ISBN 978-0-521-84802-2, MR 2165687.
• Lundberg, Erik (2007), "Almost all orbit types imply period-3", Topology and Its Applications, 154 (14): 2741–2744, doi:10.1016/j.topol.2007.05.009.
• Mortveit, Henning S.; Reidys, Christian M. (2008), An introduction to sequential dynamical systems, Universitext, New York: Springer, ISBN 978-0-387-30654-4, MR 2357144.
• Nekrashevych, Volodymyr (2008), "Symbolic dynamics and self-similar groups", Holomorphic Dynamics and Renormalization: A Volume in Honour of John Milnor's 75th Birthday, Fields Inst. Commun., vol. 53, Providence, RI: Amer. Math. Soc., pp. 25–73, MR 2477417.
• Starke, Jens; Schanz, Michael (1998), "Dynamical system approaches to combinatorial optimization", Handbook of combinatorial optimization, Vol. 2, Boston, MA: Kluwer Acad. Publ., pp. 471–524, MR 1665408.

External links

• Combinatorics of Iterated Functions: Combinatorial Dynamics & Dynamical Combinatorics
• Combinatorial dynamics at Scholarpedia