Ein Satz über die Entropie von Untermonoiden. (A theorem on the entropy of submonoids) (Q1113684)

Let X be a finite alphabet. For \(L\subseteq X^*\) let \(\rho_ L\) denote the radius of convergence of the structure generating function \[ s_ L(t):=\sum^{\infty}_{n=0}card(L\cap X^ n)\cdot t^ n \] of the language L. The entropy of L is defined as \(H_ L:=-\log_{card X} \rho_ L\). We shall prove the following proposition: Theorem: Let L be an arbitrary subset of \(X^*\). Then for every \(\epsilon >0\) there is a finite subset U of L such that \[ H_{L^*}- H_{U^*}<\epsilon. \]