Collinearity of Iterations and Real Plane Algebraic Curves

For a given meromorphic function g: C → C, we study the locus of points z that are collinear with their two iterations g(z) and g(g(z)). Such points form a collinearity curve. Generally, this curve is reducible: it has several components. Components of the curve, especially algebraic ones, are investigated. The curve´s behavior near a fixed point of the function is explored. We also look into the asymptotic structure of the curve at infinity. The components of the curve are interpretable as vertices and edges of a graph resembling a Voronoi diagram. The collinearity curve, needing a mere two iterates, may help us understanding the long-term dynamics of the original function on C, visualizing important features of the generated fractal for g and its associated Voronoi graph.