Outputs (80)

Functorial orbit counting (2009)
Journal Article
Pakapongpun, A., & Ward, T. (2009). Functorial orbit counting. Journal of integer sequences, 12, Article 09.2.4

We 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/s0002-9939-08-09649-4

We 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.

Variations in posttonsillectomy haemorrhage rates are scale invariant (2008)
Journal Article
Phillips, J., Ward, T., & Montgomery, P. (2008). Variations in posttonsillectomy haemorrhage rates are scale invariant. The Laryngoscope, 118(6), 1096-1098. https://doi.org/10.1097/mlg.0b013e3181672277

Background: Scale invariance is a property of scientific laws or objects that change in a prescribed fashion if measurements are scaled, and is often represented by a power-law relationship. Power laws suggest that events of a large magnitude will be... Read More about Variations in posttonsillectomy haemorrhage rates are scale invariant.

Uniform periodic point growth in entropy rank one (2008)
Journal Article
Miles, R., & Ward, T. (2008). Uniform periodic point growth in entropy rank one. Proceedings of the American Mathematical Society, 136(01), 359-365. https://doi.org/10.1090/s0002-9939-07-09018-1

We show that algebraic dynamical systems with entropy rank one have uniformly exponentially many periodic points in all directions.

Primes generated by recurrence sequences (2007)
Journal Article
Everest, G., Stevens, S., Tamsett, D., & Ward, T. (2007). Primes generated by recurrence sequences. The American Mathematical Monthly, 114(5), 417-431

Orbit-counting in non-hyperbolic dynamical systems (2007)
Journal Article
Everest, G., Miles, R., Stevens, S., & Ward, T. (2007). Orbit-counting in non-hyperbolic dynamical systems. Journal für die reine und angewandte Mathematik, 2007(608), 155-182. https://doi.org/10.1515/crelle.2007.056

There are well-known analogs of the prime number theorem and Mertens' theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest non-hyperbolic algebraic systems. The asymptotic behaviour of the orbit... Read More about Orbit-counting in non-hyperbolic dynamical systems.

Mixing and tight polyhedra (2006)
Book Chapter
Ward, T. (2006). Mixing and tight polyhedra. In D. Denteneer, F. den Hollander, & E. Verbitskiy (Eds.), Dynamics & stochastics (169-175). Institute of Mathematical Statistics. https://doi.org/10.1214/074921706000000194

Mixing actions of the rationals (2006)
Journal Article
Miles, R., & Ward, T. (2006). Mixing actions of the rationals. Ergodic Theory and Dynamical Systems, 26(6), 1905-1911. https://doi.org/10.1017/s0143385706000356

We study mixing properties of algebraic actions of Q^d, showing in particular that prime mixing Q^d-actions on connected groups are mixing of all orders, as is the case for Z^d-actions. This is shown using a uniform result on the solution of S-unit e... Read More about Mixing actions of the rationals.

Periodic point data detects subdynamics in entropy rank one (2006)
Journal Article
Miles, R., & Ward, T. (2006). Periodic point data detects subdynamics in entropy rank one. Ergodic Theory and Dynamical Systems, 26(6), 1913-1930. https://doi.org/10.1017/s014338570600054x

A framework for understanding the geometry of continuous actions of Z^d was developed by Boyle and Lind using the notion of expansive behaviour along lower-dimensional subspaces. For algebraic Zd-actions of entropy rank one, the expansive subdynamics... Read More about Periodic point data detects subdynamics in entropy rank one.

Primitive divisors of elliptic divisibility sequences (2006)
Journal Article
Everest, G., McLaren, G., & Ward, T. (2006). Primitive divisors of elliptic divisibility sequences. Journal of Number Theory, 118(1), 71-89. https://doi.org/10.1016/j.jnt.2005.08.002

Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of c... Read More about Primitive divisors of elliptic divisibility sequences.