📝 SummarXiv

Abstract

We study the fine structure of the parameter space of the unicritical family of algebraic correspondences $z^r + c$, where $r > 1$ is a rational exponent. Building on Tan Lei's result regarding the similarity between the Mandelbrot set and Julia sets in the quadratic family, we prove that the Julia set of the correspondence is asymptotically self-similar about every Misiurewicz point. Assuming that the transversality condition holds at a Misiurewicz parameter $a \in \mathbb{C}$, we prove that the associated Multibrot set (which coincides with the Mandelbrot set when $ r =2$) is asymptotically similar to the Julia set about $a$. We provide an algebraic proof of the transversality condition when the correspondence is represented by the semigroup $\langle z^2 +c, -z^2+c \rangle. $ For general exponents, experimental evidence supports the transversality condition, with infinitely many small copies of the Multibrot set accumulating at every Misiurewicz parameter.