The differentiability of the hairs of 𝑒𝑥𝑝(𝑍)

Abstract

We prove that the hairs of the exponential-like maps f ( z ) = λ e z f(z) = \lambda {e^z} are smooth curves. This answers affirmatively a question of Devaney and Krych. The proof is constructive in the sense that a dynamically defined C ∞ {C^\infty } parametrization is presented.