On profinite groups with automorphisms whose fixed points have countable Engel sinks

E. I. Khukhro, P. Shumyatsky

An Engel sink of an element $g$ of a group $G$ is a set $\mathcal{E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\math…