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We use the following rule of composition of automorphisms: if φ, ψ ∈ Aut(F2) and x ∈ F2 then φψ(x) = ψ(φ(x)). For x ∈ F2 denote by St(x) the stabilizer of x in Aut(F2). For a subset X of a group denote by 〈X〉 the subgroup generated by X. Denote [x, y] = x−1y−1xy, x = y−1xy. For g ∈ F2 denote by ĝ the automorphism induced by the conjugation by g: ĝ(x) = g−1xg, x ∈ F2. Let − : Aut(F2) → GL2(Z) be the homomorphism induced by the abelianization of F2. It is known that the group of inner automorphisms Inn(F2) is the kernel of this homomorphism and that

Classification of automorphisms of the free group of rank 2 by ranks of fixed-point subgroups