Outputs (80)

A repulsion motif in Diophantine equations (2011)
Journal Article
Everest, G., & Ward, T. (2011). A repulsion motif in Diophantine equations. The American Mathematical Monthly, 118(7), 584-598. https://doi.org/10.4169/amer.math.monthly.118.07.584

Problems related to the existence of integral and rational points on cubic curves date back at least to Diophantus. A significant step in the modern theory of these equations was made by Siegel, who proved that a non-singular plane cubic equation has... Read More about A repulsion motif in Diophantine equations.

A directional uniformity of periodic point distribution and mixing (2011)
Journal Article
Miles, R., & Ward, T. (2011). A directional uniformity of periodic point distribution and mixing. Discrete and Continuous Dynamical Systems - Series A, 30(4), 1181-1189. https://doi.org/10.3934/dcds.2011.30.1181

For mixing Z^d-actions generated by commuting automorphisms of a compact abelian group, we investigate the directional uniformity of the rate of periodic point distribution and mixing. When each of these automorphisms has finite entropy, it is shown... Read More about A directional uniformity of periodic point distribution and mixing.

An introduction to Number Theory (2011)
Book
Everest, G., & Ward, T. (2011). An introduction to Number Theory. Science Press, Beijing

Mertens' theorem for toral automorphisms (2011)
Journal Article
Jaidee, S., Stevens, S., & Ward, T. (2011). Mertens' theorem for toral automorphisms. Proceedings of the American Mathematical Society, 139(05), 1819-1824. https://doi.org/10.1090/s0002-9939-2010-10632-9

A dynamical Mertens' theorem for ergodic toral automorphisms with error term O(N^{-1}) is found, and the influence of resonances among the eigenvalues of unit modulus is examined. Examples are found with many more, and with many fewer, periodic orbit... Read More about Mertens' theorem for toral automorphisms.

A dichotomy in orbit growth for commuting automorphisms (2010)
Journal Article
Miles, R., & Ward, T. (2010). A dichotomy in orbit growth for commuting automorphisms. Journal of the London Mathematical Society, 81(3), 715-726. https://doi.org/10.1112/jlms/jdq010

We consider asymptotic orbit-counting problems for certain expansive actions by commuting automorphisms of compact groups. A dichotomy is found between systems with asymptotically more periodic orbits than the topological entropy predicts, and those... Read More about A dichotomy in orbit growth for commuting automorphisms.

Dirichlet series for finite combinatorial rank dynamics (2010)
Journal Article
Everest, G., Miles, R., Stevens, S., & Ward, T. (2010). Dirichlet series for finite combinatorial rank dynamics. Transactions of the American Mathematical Society, 362(01), 199-227. https://doi.org/10.1090/s0002-9947-09-04962-9

We introduce a class of group endomorphisms -- those of finite combinatorial rank -- exhibiting slow orbit growth. An associated Dirichlet series is used to obtain an exact orbit counting formula, and in the connected case this series is shown to hav... Read More about Dirichlet series for finite combinatorial rank dynamics.

Planar dynamical systems with pure Lebesgue diffraction spectrum (2010)
Journal Article
Baake, M., & Ward, T. (2010). Planar dynamical systems with pure Lebesgue diffraction spectrum. Journal of Statistical Physics, 140(1), 90-102. https://doi.org/10.1007/s10955-010-9984-x

We examine the diffraction properties of lattice dynamical systems of algebraic origin. It is well-known that diverse dynamical properties occur within this class. These include different orders of mixing (or higher-order correlations), the presence... Read More about Planar dynamical systems with pure Lebesgue diffraction spectrum.

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

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.2009.24.613

We 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-y

We 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.