Dynamics of a certain sequence of powers

Abstract

For any nonzero complex number z we define a sequence a1(z)=z, a2(z)=za1(z),…,an+1(z)=zan(z), n∈ℕ. We attempt to describe the set of these z for which the sequence {an(z)} is convergent. While it is almost impossible to characterize this convergence set in the complex plane 𝒞, we achieved it for positive reals. We also discussed some connection to the Euler's functional equation.

Keywords

• Power sequence
• dynamics
• convergence.