All 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

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.

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.

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.

Orbit counting with an isometric direction (2005)
Book Chapter
Everest, G., Stangoe, V., & Ward, T. (2005). Orbit counting with an isometric direction. In S. Kolyada, Y. Manin, & T. Ward (Eds.), Algebraic and topological dynamics (293-302). American Mathematical Society. https://doi.org/10.1090/conm/385/07202

Algebraic and Topological Dynamics (2005)
Presentation / Conference Contribution
Kolyada, S., Manin, Y., & Ward, T. (2005). Algebraic and Topological Dynamics.

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.

An Introduction to Number Theory (2005)
Book
Everest, G., & Ward, T. (2005). An Introduction to Number Theory. Springer Verlag

An Introduction to Number Theory provides an introduction to the main streams of number theory. Starting with the unique factorization property of the integers, the theme of factorization is revisited several times throughout the book to illustrate h... Read More about An Introduction to Number Theory.

Ergodentheorie: von Planetenbahnen zur Zahlentheorie (2005)
Book
Ward, T. (Ed.). (2005). Ergodentheorie: von Planetenbahnen zur Zahlentheorie. Max Planck Gesellschaft

Die Theorie der dynamischen Systeme liefert eine wirkungsvolle Methode um unterschiedliche Probleme von Planetenbahnen bis hin zur klassischen Zahlentheorie zu untersuchen. Indem Einsiedler, Katok und Lindenstrauss in ihren neuesten Arbeiten Methoden... Read More about Ergodentheorie: von Planetenbahnen zur Zahlentheorie.

Group automorphisms with few and with many periodic points (2005)
Journal Article
Ward, T. (2005). Group automorphisms with few and with many periodic points. Proceedings of the American Mathematical Society, 133(01), 91-96. https://doi.org/10.1090/s0002-9939-04-07626-9

Morphic heights and periodic points (2004)
Book Chapter
Einsiedler, M., Everest, G., & Ward, T. (2004). Morphic heights and periodic points. In D. Chudnovsky, G. Chudnovsky, & M. Nathanson (Eds.), Number theory : New York seminar 2003 (167-177). Springer Verlag

An approach to the calculation of local canonical morphic heights is described, motivated by the analogy between the classical height in Diophantine geometry and entropy in algebraic dynamics. We consider cases where the local morphic height is expre... Read More about Morphic heights and periodic points.

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.