Classification of connected palette diagrams without area and moment to find relations of formal diffeomorphisms

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For a polygonal path γ consisting of unit vectors in x-direction and y-direction in the plane R^2, a relation W_γ*(f, g)=id of f, g is defined, where id denotes the identity map of C. Some sufficient conditions of γ so that W_γ*(f, g)=id admits solutions of non commuting formal diffeomorphisms tangent to the identity have already been obtained in [4]. In this paper, we define a palette diagram and classify all connected palette diagrams without area and moment consisting of four unit weighted squares into 4 types to find γ so that W_γ*(f, g)=id admits solutions of non commuting formal diffeomorphisms tangent to the identity among those diagrams. We also give some concrete examples of relations of two formal diffeomorphisms.

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