An Entire Function Analogue of Smale's 12th Problem on Centralizers

In 1998, Stephen Smale proposed 18 mathematical problems for the 21st century. Problem 12 asks whether a diffeomorphism of a compact manifold can be approximated by one whose centralizer is trivial. This was solved in the $C^1$ topology by Bonatti, Crovisier, and Wilkinson (2009), but remains open for higher regularities. In this paper, we formulate and resolve an analogue of Smale's 12th problem for entire functions on the complex plane. Specifically, we prove that the set of non-constant entire functions with trivial centralizer is dense in the space of all entire functions equipped with the topology of locally uniform convergence.