Let $(M^{2n},J)$ be an almost complex manifold and $(\Sigma,j)$ a closed Riemann surface. Suppose $u: \Sigma \to M$ is a simple, nonconstant, $J$-holomorphic curve. Can we prove that the set $$ Z:=\{z\in \Sigma \mid |u^{-1}(u(z))|>1\} $$ is finite?

Note: It follows from Proposition 2.5.1 in McDuff-Salamon's book "J-holomorphic Curves and Symplectic Topology" that $Z$ is countable.