Hereditarily non uniformly perfect non-autonomous Julia sets
Date of Original Version
Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in  who gave several examples of such sets based on Cantor set-like constructions using nested intervals. We exhibit a class of examples in non-autonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary. In particular, we give a sharp criterion for when Julia sets from our class will be HNUP and we show that the maximum possible Hausdorff dimension of 1 for these Julia sets can be attained. The proof of the latter considers the Julia set as the limit set of a non-autonomous conformal iterated function system and we calculate the Hausdorff dimension using a version of Bowen’s formula given in the paper by Rempe-Gillen and Urbánski .
Discrete and Continuous Dynamical Systems- Series A
Comerford, Mark, Rich Stankewitz, and Hiroki Sumi. "Hereditarily non uniformly perfect non-autonomous Julia sets." Discrete and Continuous Dynamical Systems- Series A 40, 1 (2020): 33-46. doi:10.3934/dcds.2020002.