Glasnik Matematicki, Vol. 61, No. 1 (2026), 77-115. \( \)

MONODROMY THROUGH NARROW BIFURCATION LOCUS OF THE MANDELBROT SET

Hyungryul Baik and Juhun Baik

Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, South Korea
e-mail:hrbaik@kaist.ac.kr

Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, South Korea
e-mail:jhbaik@kaist.ac.kr

Abstract.   We study the behavior of the itinerary sequence of each point of the Julia set of \(z\mapsto z^2 + c\) when the parameter \(c\) in the shift locus is allowed to pass through points in the bifurcation locus, which we call narrow. We first show the combinatorial and geometric properties of narrow characteristic arcs. We also show how the itinerary sequence changes in an algorithmic way by using the lamination models proposed by Keller [13]. The algorithm is recently generalized to all bifurcation points by Ishii and Richards [11]. Finally, we found an equivalence relation on the set of \(0\)-\(1\) sequences such that the changing rule is a shift invariant up to the equivalence relation. This generalizes Atela's works ([1], [2]), which dealt with the special case of the generalized rabbit polynomials.

2020 Mathematics Subject Classification.   37F46, 37F20, 37F10

Key words and phrases.   bifurcation locus, itinerary sequence, kneading sequence, Mandelbrot set, invariant circle lamination.

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https://doi.org/10.3336/gm.61.1.04

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