On the topological structure of the space of ℤᵈ-subshifts
Date(s) : 10/02/2026 iCal
11h00 - 12h00
The space 𝓢ᵈ of ℤᵈ-subshifts, d≥1, endowed with the Hausdorff topology, provides a natural framework for the study of generic properties, as it admits an interpretation in terms of languages. For one-dimensional systems several generic properties have been characterised and, in a recent work, Pavlov and Schmieding have finished painting the picture: they characterised isolated points and showed that they completely account for generic systems and the non-perfect structure of 𝓢¹, as the Cantor-Bendixson rank of the space is 1. For higher-dimensional systems the picture has yet to be finished.
In this talk, we revisit a previous joint work with Silvère Gangloff, where we characterised isolated points in 𝓢ᵈ in terms of maximal subsystems –this characterisation allows, in particular, to recover the one-dimensional characterisation. We discuss the existence of a non-isolated subshift that can only be approximated by isolated systems, making it isolated in the derived set of 𝓢ᵈ, and then iterate this construction to produce analogous systems at every finite level of the Cantor-Bendixson process, showing that the Cantor-Bendixson rank of 𝓢² is infinite. Finally, we present our latest progress on this topic: we propose a complementary approach based on the lattice of subsystems and its order-theoretic structure. This new perspective allows us to shed light on the structure of isolated points in the derivative of 𝓢ᵈ.
joint with Silvère Gangloff
Emplacement
I2M Luminy - TPR2, Salle de Séminaire 304-306 (3ème étage)
Catégories
