Leinster's categorical theory of self-similarity

In Tom Leinster's 2010 paper ``A general theory of self-similarity'', a comprehensive theory of equational systems is developed, and self-similar topological spaces are characterized as certain coalgebras, arising as universal solutions of equational systems. The theory includes techniques for determining whether a universal solution exists, for constructing it explicitly when it does, and for checking whether a candidate coalgebra is indeed the universal solution of a given equational system. We present a survey of Leinster's theory, together with some considerations on a conjecture of the author concerning the Julia sets of rational maps of the Riemann sphere.