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

Let $F_n$ be a free group of rank n generated by $x_1,\dots ,x_n$. In this paper we discuss three algorithmic problems related to automorphisms of $F_2$.A word $u=u(x_1,\dots ,x_n)$ of $F_n$ is called positive if no negative exponents of $x_i$ occur in u. A word u in $F_n$ is called potentially positive if $?(u)$ is positive for some automorphism ? of $F_n$. We prove that there is an algorithm to decide whether or not a given word in $F_2$ is potentially positive, which gives an affirmative solution to problem F34a in G. Baumslag, A.G. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, second ed., in: Contemp. Math., vol. 296, 2002, pp. 1–38, online version: http://www.grouptheory.info] for the case of $F_2$.Two elements u and v in $F_n$ are said to be boundedly translation equivalent if the ratio of the cyclic lengths of $?(u)$ and $?(v)$ is bounded away from 0 and from ∞ for every automorphism ? of $F_n$. We provide an algorithm to determine whether or not two given elements of $F_2$ are boundedly translation equivalent, thus answering question F38c in the online version of G. Baumslag, A.G. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, second ed., in: Contemp. Math., vol. 296, 2002, pp. 1–38, online version: http://www.grouptheory.info] for the case of $F_2$.We also provide an algorithm to decide whether or not a given finitely generated subgroup of $F_2$ is the fixed point group of some automorphism of $F_2$, which settles problem F1b in G. Baumslag, A.G. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, second ed., in: Contemp. Math., vol. 296, 2002, pp. 1–38, online version: http://www.grouptheory.info] in the affirmative for the case of $F_2$.