36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

"Substitutions on compact alphabets"

Rust, Dan

One dimensional substitutions and their tiling spaces are classical objects in tiling theory 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 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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