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

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