Shift-preserving maps on $\omega^*$

Abstract

The shift map $\sigma$ on $\omega^*$ is the continuous self-map of $\omega^*$ induced by the function $n \mapsto n+1$ on $\omega$. Given a compact Hausdorff space $X$ and a continuous function $f: X \rightarrow X$, we say that $(X,f)$ is a quotient of $(\omega^*,\sigma)$ whenever there is a continuous surjection $Q: \omega^* \to X$ 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 $(\omega^*,\sigma)$ 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 $(\omega^*,\sigma)$ and implies, for example, that $(\omega^*,\sigma^{-1})$ is a quotient of $(\omega^*,\sigma)$. 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 $(\omega^*,\sigma^{-1})$ is not a quotient of $(\omega^*,\sigma)$.