Composite rational functions in view of lacunarity

Let f be a rational function that is the composite of two rational functions g,h. In this talk we discuss the question of what can be said about the composition factors g,h when we assume that f is lacunary e.g. in the sense that the number of terms in a given representation of f as quotient of two polynomials is fixed (but also other notions of lacunarity are of interest and will be studied). The results support and quantify the intuitive expectation that rational operations of large degree tend to destroy the lacunarity.