Title: Morphisms in symbolic dynamics: Isomorphism, embedding, factors, and retractions.
Abstract:
"Symbolic dynamics" is concerned with the study of sequences of "symbols" from the point of view of topological dynamics: The space of $A^\mathbb{Z}$ of (bi-infinite) sequences over a finite (discrete) "alphabet" $A$, becomes a compact metrizable space when endued with the product topology, and the shift map is a self-homeomorphism.
A subshift is a closed, shift-invariant subset of $A^{\mathbb{Z}$.
A subshift of finite type is a space of sequences defined by a finite number of local, shift-invariant constraints (eg. certain symbols are not allowed to follow others). In multidimensional symbolic dynamics, the objects are "multidimensional" (symbolic) sequences (eg. functions from $\mathbb{Z}^d$ or more general countable discrete groups).
From a category theory point of view, subshifts are the objects, where the morphisms are continuous, shift-commuting continuous maps.
As in many other areas in mathematics, many of the fundamental questions in symbolic dynamics take the following general form: When are two subshifts isomorphic? When does one subshift embed in another subshift? When does one subshift factor onto another?
The isomorphism problem in symbolic dynamics is long-standing and notoriously open, already for one-dimensional subshifts of finite type. The embedding and factoring problems for one-dimensional (topologically mixing) subshifts of finite type are (mostly) addressed by Krieger's embedding theorem and Boyle’s factor theorems. These are both classical theorems from the 1980's.
Krieger's embedding theorem has found surprising applications in other areas of mathematics.
In my talk, I will explain Krieger's embedding theorem together with a multidimensional generalization that I obtained recently.
This new multidimensional generalization of Krieger's utilizes the notion of a retraction, that seems to be significant also in the context of symbolic dynamics.
Livestream/Recording Link: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=c9086bb3-f159-4cf6-867b-b124006dc2af