On several problems about automorphisms of the free group of rank two

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Let Fₙ be a free group of rank n. In this paper we discuss three algorithmic problems related to automorphisms of F₂. A word u of Fₙ is called positive if u does not have negative exponents. A word u in Fₙ is called potentially positive if ϕ(u) is positive for some automorphism ϕ of Fₙ. We prove that there is an algorithm to decide whether or not a given word in F₂ is potentially positive, which gives an affirmative solution to problem F34a in [1] for the case of F₂. Two elements u and v in Fₙ are said to be boundedly translation equivalent if the ratio of the cyclic lengths of ϕ(u) and ϕ(v) is bounded away from 0 and from ∞ for every automorphism ϕ of Fₙ. We provide an algorithm to determine whether or not two given elements of F₂ are boundedly translation equivalent, thus answering question F38c in the online version of [1] for the case of F₂. We further prove that there exists an algorithm to decide whether or not a given finitely generated subgroup of F₂ is the fixed point group of some automorphism of F₂, which settles problem F1b in [1] in the affirmative for the case of F₂.