Algebraic Invariants in Symbolic Dynamics

In the EPSRC grant EP/V007459/1, Reem Yassawi studies ways to compute the automorphism group and the Ellis semigroup for low complexity symbolic dynamical systems, including those defined by combinatorial rules called substitutions.

Symbolic dynamics is a branch of dynamical systems theory where where both space and time are discrete. It originated in the study of the recurrence properties of geodesic flows on surfaces. Today it is a wide area, attracting researchers in and ergodic theory and topological dynamics, encompassing the study of both positive entropy systems at one end, and small, self-similar dynamical systems at the other.

We can classify dynamical systems by defining and computing invariants for them. For example, the entropy of a system is a numerical invariant which captures the extent to which a system is unpredictable. We can also define more involved algebraic invariants, such as the automorphism group of a dynamical system, which encodes its symmetries, and the Ellis semigroup, which is a certain compactification of the dynamics (group) acting on the space. For some systems, such as group rotations, there are good descriptions of the Ellis semigroup. For this project we aim to build on previous work where the Ellis semigroup for certain dynamical systems can be computed and better understood.

An example of a substitution rule (left), coding (below right), and an element of the phase space whose Ellis semigroup we would like to understand.