Shift-preserving maps on ${\omega }^{\ast }$

Abstract

The shift map $\sigma$ on ${\omega }^{\ast }$ is the continuous self-map of ${\omega }^{\ast }$ induced by the function $n\mapsto n+1$ on $\omega$. Given a compact Hausdorff space $X$ and a continuous function $f:X\to X$, we say that $(X,f)$ is a quotient of $({\omega }^{\ast },\sigma )$ whenever there is a continuous surjection $Q:{\omega }^{\ast }\to X$ such that $Q\circ \sigma =f\circ Q$. Our main theorem states that if the weight of $X$ is at most ${\aleph }_1$, then $(X,f)$ is a quotient of $({\omega }^{\ast },\sigma )$ if and only if $f$ is weakly incompressible (which means that no nontrivial open $U\subseteq X$ has $f(\bar{U})\subseteq U$). Under CH, this gives a complete characterization of the quotients of $({\omega }^{\ast },\sigma )$ and implies, for example, that $({\omega }^{\ast },{\sigma }^{-1})$ is a quotient of $({\omega }^{\ast },\sigma )$. In the language of topological dynamics, our theorem states that a dynamical system of weight ${\aleph }_1$ is an abstract $\omega$-limit set if and only if it is weakly incompressible. We complement these results by proving $(1)$ our main theorem remains true when ${\aleph }_1$ is replaced by any $\kappa <\mathfrak{p}$, $(2)$ consistently, the theorem becomes false if we replace ${\aleph }_1$ by ${\aleph }_2$, and $(3)$ OCA+MA implies that $({\omega }^{\ast },{\sigma }^{-1})$ is not a quotient of $({\omega }^{\ast },\sigma )$.