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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 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.

Entropy geometry and disjointness for zero-dimensional algebraic actions (2005)
Journal Article
Einsiedler, M., & Ward, T. (2005). Entropy geometry and disjointness for zero-dimensional algebraic actions. Journal für die reine und angewandte Mathematik, 584, 195-214. https://doi.org/10.1515/crll.2005.2005.584.195

We show that many algebraic actions of higher-rank abelian groups on zero-dimensional compact abelian groups are mutually disjoint. The proofs exploit differences in the entropy geometry arising from subdynamics and a form of Abramov-Rokhlin formula... Read More about Entropy geometry and disjointness for zero-dimensional algebraic actions.

Finite entropy characterizes topological rigidity on connected groups (2005)
Journal Article
Bhattacharya, S., & Ward, T. (2005). Finite entropy characterizes topological rigidity on connected groups. Ergodic Theory and Dynamical Systems, 25(2), 365-373. https://doi.org/10.1017/s0143385704000501

Let X, Y be mixing connected algebraic dynamical systems with the Descending Chain Condition. We show that every equivariant continuous map from X to Y is affine (that is, Y is topologically rigid) if and only if the system Y has finite topological e... Read More about Finite entropy characterizes topological rigidity on connected groups.

Isomorphism rigidity in entropy rank two (2005)
Journal Article
Einsiedler, M., & Ward, T. (2005). Isomorphism rigidity in entropy rank two. Israel Journal of Mathematics, 147(1), 269-284. https://doi.org/10.1007/bf02785368

We study the rigidity properties of a class of algebraic Z^3-actions with entropy rank two. For this class, conditions are found which force an invariant measure to be the Haar measure on an affine subset. This is applied to show isomorphism rigidity... Read More about Isomorphism rigidity in entropy rank two.

Asymptotic geometry of non-mixing sequences (2003)
Journal Article
Einsiedler, M., & Ward, T. (2003). Asymptotic geometry of non-mixing sequences. Ergodic Theory and Dynamical Systems, 23(1), 75-85. https://doi.org/10.1017/s0143385702000950

The exact order of mixing for zero-dimensional algebraic dynamical systems is not entirely understood. Here we use valuations in function fields to exhibit an asymptotic shape in non-mixing sequences for algebraic Z^2-actions. This gives a relationsh... Read More about Asymptotic geometry of non-mixing sequences.

Integer sequences and periodic points (2002)
Journal Article
Everest, G., van der Poorten, A., Puri, Y., & Ward, T. (2002). Integer sequences and periodic points. Journal of integer sequences, 5, Article 02.2.3

Arithmetic properties of integer sequences counting periodic points are studied, and applied to the case of linear recurrence sequences, Bernoulli numerators, and Bernoulli denominators.