Topological Entropy of Regular Chaotic Quadratic Maps

The complex quadratic map as iterated complex curvature is discussed with respect to topological entropy and physical implications. The general quadratic map is a Hermite-Tschirnhaus relation of cubic invariants which can be casted into a binary linear substitution. It serves as a quadratic transformation of the elliptic integral of Friedmann equations. For invariant substitutions of holomorphic functions, a charge definition is proposed for the Riemann Xi-function and the Dirichlet L-function in form of a two-dimensional Poisson equation.