Expansivity properties and rigidity for non-recurrent exponential maps

emphsmall We will set up and prove a rigidity statement for an exponential map $f_c(z)=e^z+c$ such that the singular value $c$ is combinatorially non-recurrent, non-escaping and belongs to the Julia set. We will also prove a theorem about expansivity of the postsingular set in the case that the singular value is non-recurrent, generalizing results of Rempe and van Strien to the case in which the postsingular set is not bounded.