A right Engel sink of an element g of a group G is a set R(g)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathscr {R}}}(g)$$\end{document} such that for every x∈G\documentclass[12pt]{minimal}
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\begin{document}$$x\in G$$\end{document} all sufficiently long commutators [...[[g,x],x],⋯,x]\documentclass[12pt]{minimal}
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\begin{document}$$[...[[g,x],x],\dots ,x]$$\end{document} belong to R(g)\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {R}}(g)$$\end{document}. (Thus, g is a right Engel element precisely when we can choose R(g)={1}\documentclass[12pt]{minimal}
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\begin{document}$${{\mathscr {R}}}(g)=\{ 1\}$$\end{document}.) We prove that if a profinite group G admits a coprime automorphism φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document} of prime order such that every fixed point of φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document} has a finite right Engel sink, then G has an open locally nilpotent subgroup. A left Engel sink of an element g of a group G is a set E(g)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathscr {E}}}(g)$$\end{document} such that for every x∈G\documentclass[12pt]{minimal}
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\begin{document}$$x\in G$$\end{document} all sufficiently long commutators [...[[x,g],g],⋯,g]\documentclass[12pt]{minimal}
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\begin{document}$$[...[[x,g],g],\dots ,g]$$\end{document} belong to E(g)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathscr {E}}}(g)$$\end{document}. (Thus, g is a left Engel element precisely when we can choose E(g)={1}\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {E}}(g)=\{ 1\}$$\end{document}.) We prove that if a profinite group G admits a coprime automorphism φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document} of prime order such that every fixed point of φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document} has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.
