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Cylinders, multi-cylinders and the induced action of $Aut(F_n)$

A cylinder $C^1_u$ is the set of infinite words with fixed prefix $u$. A double-cylinder $C^2_{[1,u]}$ is "the same" for bi-infinite words. We show that for every word $u$ and any automorphism $φ$ of the free group $F$ the image $φ(C^1_u)$ is a finite union of cylinders. The analogous statement is true for double cylinders. We give (a) an algorithm, and (b) a precise formula which allows one to determine this finite union of cylinders.