Summary. Periodic behaviour in dynamical systems (for example, the way in which planets orbit the Sun regularly) is known to be important in understanding how physical systems evolve over time. In mathematics we study abstract models of dynamical systems. This project deals with one aspect of a kind of dynamical system in which evolution over time is replaced by a higher-dimensional action. Thus it makes sense to let our abstract model system evolve (like running a planetary orbit forward for thousands of years) in any one of infinitely many different directions. From studying examples, we think we have some ideas about how periodic orbits in our abstract system (points that return to themselves exactly under the dynamics) behave in different directions of 'time'. This project is designed to give a complete picture of how this periodic behaviour changes as the direction is changed. We expect that the results from this project will be used as part of the research efforts by many different people who study the same type of abstract dynamical systems. These abstract dynamical systems are of interest in themselves, and as testing grounds for developing what we think will be true of more complicated systems. They also have many connections to other parts of mathematics.

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