Commutativity of flows and injectivity of nonsingular mappings (Q2724128)

The author studies the relationship between Jacobian maps and dynamic properties of suitable plane Hamiltonian systems. Moreover, the author shows that in order to prove the Jacobian conjecture in the real plane it is sufficient to prove that, for every polynomial Jacobian map \(\Lambda\), there exists an integer \(n_\Lambda\) and a real constant \(h_\Lambda\) such that the following inequality holds, outside a suitable compact subset of the plane: NEWLINE\[NEWLINE(fg_y-gf_y)^2 +(-fg_x+ gf_x)^2\leq h_\Lambda (x^2+y^2) (f^2+g^2)^{2 n_\Lambda}d,NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\begin{cases} \dot x={fy \over d}\\ \dot y=-{f_x\over d} \end{cases} \text{ and }\begin{cases} \dot x=g_y/d\\ \dot y=-{g_x\over d}\end{cases};\quad z \mapsto\Lambda (z)=\bigl(f(z), g(z)\bigr),\;z=(x,y).NEWLINE\]