2025/07/17 by Mark Comerford, Comerford, Mark, Hiroki Sumi +1 · 1 citation
Computer Science · #Cellular Automata and Applications #Graph Theory and Algorithms #Advanced Data Storage Technologies
paper · pdf · doi:10.48550/arxiv.2507.12929
Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in \citeSSS who gave several examples of such sets based on Cantor set-like constructions using nested intervals. For non-autonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary, the Julia set is uniformly perfect for all sequences with suitably bounded coefficients, while Comerford, Stankewitz and Sumi showed in \citeCSS that for certain sequences of polynomials with unbounded coefficients, it is possible to have Julia sets which are HNUP. In this manuscript we give an example of a non-autonomous polynomial sequences whose Julia sets lie in between these two extremes in that they are not uniformly perfect, but also not HNUP. In addition we show that these Julia sets can be expressed as a `thick-thin' decomposition consisting of a \mathrm Fσ subset which is a countable union of uniformly perfect sets and a \mathrm Gδ subset which is HNUP.