The fixed-point criterion equivalent to the Jacobian conjecture

Let $P:\mathbb C^2\rightarrow\mathbb C^2$ satisfy $J(P)\equiv 1$. Suppose there exists an algebraic function defined on some open set ${\cal U}$ such that $P\circ Q=P$, and suppose that $Q$ has finite order. The fixed-point criterion equivalent to the Jacobian conjecture. The map $Q$ has a fixed point. The source presents this fixed-point statement as equivalent to the Jacobian conjecture; the paper's surrounding discussion indicates that this formulation is proposed as a route to proving the conjecture, while the general Jacobian conjecture remains open.