Tame or wild Toeplitz shifts
Ergodic Theory and Dynamical Systems Seminar
22nd October 2020, 4:30 pm – 5:30 pm
Online, on zoom (to participate, email the organisers for a link),
The Ellis semigroup $E(X,T)$ of a topological dynamical system is defined to be the compactification of the action $T$ in the topology of pointwise convergence on the space of all functions $X^X$. Tameness is a concept whose roots date back to Rosenthal’s $\ell^1$ embedding theorem, which says that if a sequence in $\ell^1$ does not have a weakly Cauchy subsequence, then it must be the sequence of unit vectors in $\ell^1$. Koehler linked the concept of tameness to the Ellis semigroup. A system is tame if its Ellis semigroup has size at most the continuum. Non-tame systems are very far from tame, as they must contain a copy of $\beta \mathbb N$, the Stone-Cech compactification of $\mathbb N$.
Since then, the dynamics community has investigated the question of which systems are tame. In this talk I will give a brief exposition of these results, and talk about my recent work with Gabriel Fuhrmann and Johannes Kellendonk, where we study tameness, or otherwise, of Toeplitz shifts.