English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Given d ≥ 2 consider the family of polynomials Pc(z) = z + c for c ∈ C. Denote by Jc the Julia set of Pc and letMd = {c | Jc is connected} be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters c0 ∈ ∂Md: those for which the critical point 0 is not recurrent by Pc0 and without parabolic cycles. The Hausdorff dimension of Jc, denoted by HD(Jc), does not depend continuously on c at such c0 ∈ ∂Md; on the other hand the function c 7→ HD(Jc) is analytic in C − Md. Our first result asserts that there is still some continuity of the Hausdorff dimension if one approaches c0 in a “good” way: there is C = C(c0) > 0 such that for a sequence cn → c0, if dist(cn,Md) ≥ C|cn − c0|, then HD(Jcn)→ HD(Jc0). To prove this we use the fact that Md and Jc0 are similar near c0. In fact we prove that the biholomorphism ψ : C−Jc0 → C−Md tangent to the identity at infinity is conformal at c0: there is λ 6= 0 such that ψ(w) = c0 + λ(w − c0) +O(|w − c0|) for w 6∈ Jc0 . This implies that the local structures ofMd and Jc0 at c0 are similar. The fact that λ 6= 0 is related to a transversality phenomenon that is well known for Misiurewicz parameters and that we extend to the semihyperbolic case. We also prove that for some C > 0, dH(Jc, Jc0) ≤ C|c− c0| and dH(Kc, Jc0) ≤ C|c− c0|, where dH denotes the Hausdorff distance.

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets