紀要論文

紀要論文

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Osaka Journal of Mathematics. 2025, 62(4), p. 609-632

CONNECTIVITY OF JULIA SETS OF HÉNON MAPS NEAR THE BOUNDARY OF THE MANDELBROT SET

Yagi, Takumi

Departments of Mathematics, The University of Osaka and Osaka Metropolitan University

英語

text

2025-10

In this paper, we prove some new connectivity of the Julia sets J of the complex Hé non maps H(x, y) = (x²+c+ay, ax) with sufficiently small |a|. We investigate the connectivity of J for the parameters near the boundary of the Mandelbrot set. We first give some conditions related to the connectivity of J for sufficiently small |a|, which are useful for considering the connectivity of J for the parameters near the boundary of the Mandelbrot set. We consider a perturbation {Hₐ,_λₜ}_a∈D_δ₀,0≤t<δ₀ of dissipative semi-parabolic Hénon maps Hₐ,_λ₀ such that det DHₐ,_λₜ = −a² and Hₐ,_λₜ has a fixed point qₐ,_λₜ for which (DHₐ,_λₜ )_qₐ,_λₜ has an eigenvalue λₜ . Assume that λₜ → λ₀ = exp(2πim/l) ∈ ∂D as t → 0 and λ^l_t can be represented by exp(Lₜ + iθₜ) with Lₜ ≠ 0 for 0 < t < δ₀. We prove that if θₜ = O(Lₜ ), then the Julia sets Jₐ,_λₜ for a ∈ D_δ₀ , 0 < t < δ₀ are connected by using the conditions above.

10.18910/102980

102980

CONNECTIVITY OF JULIA SETS OF HÉNON MAPS NEAR THE BOUNDARY OF THE MANDELBROT SET

Yagi, Takumi

Departments of Mathematics, The University of Osaka and Osaka Metropolitan University

英語

2025-10

Osaka Journal of Mathematics

62

4

609

632

https://hdl.handle.net/11094/102980

10.18910/102980

https://doi.org/10.18910/102980