Ergodic theory: interactions with combinatorics and number theory
(2009)
Book Chapter
Ward, T. (2009). Ergodic theory: interactions with combinatorics and number theory. In R. Meyers (Ed.), Encyclopedia of complexity and systems science (3040-3053). Springer Verlag. https://doi.org/10.1007/springerreference_60353
Outputs (5)
Markov partitions reflecting the geometry of x2,x3 (2009)
Journal Article
Ward, T., & Yayama, Y. (2009). Markov partitions reflecting the geometry of x2,x3. Discrete and Continuous Dynamical Systems - Series A, 24(2), 613-624. https://doi.org/10.3934/dcds.2009.24.613We give an explicit geometric description of the $\times2,\times3$ system, and use his to study a uniform family of Markov partitions related to those of Wilson and Abramov. The behaviour of these partitions is stable across expansive cones and trans... Read More about Markov partitions reflecting the geometry of x2,x3.
The continuing story of zeta (2009)
Journal Article
Everest, G., Röttger, C., & Ward, T. (2009). The continuing story of zeta. Mathematical Intelligencer, 31(3), 13-17. https://doi.org/10.1007/s00283-009-9053-yWe show how the Binomial Theorem can be used to continue the Riemann Zeta Function to the left hand half-plane. This method yields the explicit values of the function at non-positive integers in terms of the Bernoulli numbers.
Functorial orbit counting (2009)
Journal Article
Pakapongpun, A., & Ward, T. (2009). Functorial orbit counting. Journal of integer sequences, 12, Article 09.2.4We study the functorial and growth properties of closed orbits for maps. By viewing an arbitrary sequence as the orbit-counting function for a map, iterates and Cartesian products of maps define new transformations between integer sequences. An orbit... Read More about Functorial orbit counting.
Orbit-counting for nilpotent group shifts (2009)
Journal Article
Miles, R., & Ward, T. (2009). Orbit-counting for nilpotent group shifts. Proceedings of the American Mathematical Society, 137(04), 1499-1507. https://doi.org/10.1090/s0002-9939-08-09649-4We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens' theorem for the full $G$-shift for a finitely-generated torsion-free nilpotent group $G$. Using bounds for the M{\"o}bius function on the lattice of subgroups o... Read More about Orbit-counting for nilpotent group shifts.