Document Type
Honors Project - Open Access
Abstract
We consider the basilica Julia set of the quadratic polynomial P (z) = z^2 - 1, with its successive graph approximations defined in terms of the external ray parametrization of the set. Following the model of Kigami and later Strichartz, we exploit these graph approximations to define derivatives of functions defined on the fractal, an endeavor complicated by asymmetric neighborhood behaviors at approximated vertex points across levels, and by the topology of these vertices. We hence differentiate even and odd levels of approximations of the Julia set and construct, accordingly, normal derivatives corresponding to each level category at the vertices, given their assigned ray names. We also discuss how a localized harmonic function serves as the tangent line, from which local linear approximation near vertices are obtained.
Recommended Citation
Xia, Xintan and Flock, Taryn, "First Order Approximation on the Basilica Julia Set" (2023). Mathematics, Statistics, and Computer Science Honors Projects. 83.
https://digitalcommons.macalester.edu/mathcs_honors/83
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