Symbolic Dynamics and Aperiodic Order

Symbolic dynamics is the study of dynamical systems where both space and time are discrete. At the OU, researchers are particularly interested in symbolic dynamical systems which possess self-similarity properties, and which often have zero entropy. These include substitution dynamical systems and various generalisations, such as S-adic systems, random substitution systems and Bratteli-Vershik systems. A current EPSRC-funded project involves studying the algebraic invariants of these systems.

Aperiodic order is concerned with the investigation of spatial structures, such as point sets or tilings, which are ordered, yet without any translational symmetries. Their study was reinforced by the discovery of aperiodic crystals (quasicrystals). Tilings generate dynamical systems whose spectral study links back to the diffraction properties of their physical manifestations. One current EPSRC-funded project considers spectral properties of inflation tilings. In addition, investigators collaborate with the School of Engineering and Innovation on an EPSRC New Horizons grant to model mechanical properties of three-dimensional physical tilings.

Photograph of the Penrose tiling at the Mathematical Institute, University of Oxford. The particular decoration of this tiling was designed by Sir Roger Penrose himself

MSc students interested in learning more about this area of research can consider taking the topic ATSD (Aperiodic tilings and symbolic dynamics) in our Masters dissertation module M840.