Boundary Fixed Points and Infinitesimal Generators

Abstract

In this chapter, we continue the analysis of the boundary behavior of semigroups, and we concentrate on boundary fixed points. We show that if a positive iterate of a semigroup has a boundary fixed point (in the sense of non-tangential limit), then such a point is indeed fixed for all the iterates of the semigroup. Moreover, the boundary dilation coefficients of the semigroup at a boundary fixed point are either identically infinity or they are of the form $e^{-\lambda t}$ for some $\lambda \in \mathbb R$. The latter is the case of boundary regular fixed points. We prove that there is a one-to-one correspondence between boundary regular fixed points and boundary regular critical points of infinitesimal generators.