### Substitutions on compact alphabet

Ergodic Theory and Dynamical Systems Seminar

28th October 2021, 2:00 pm – 3:00 pm

Fry Building, 2.04 (TBC)

Substitutions and their subshifts are classical objects in symbolic dynamics representing some of the most well-studied and 'simple' aperiodic systems. Classically they are defined on finite alphabets, but it has recently become clear that a systematic study of substitutions on infinite alphabets is needed. I'll introduce natural generalisations (for compact Hausdorff alphabets) of classical concepts like legal words, repetitivity, primitivity, etc., and report on new progress in an attempt to characterise unique ergodicity of these systems, where surprisingly, primitivity is not sufficient. As Perron-Frobenius theory fails in infinite dimensions, more sophisticated technology from the theory of positive (quasicompact) operators is employed. There are still lots of open questions, and so a ground-level introduction to these systems will hopefully be approachable and stimulating.

This is joint work with Neil Manibo and Jamie Walton.

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