Semi-parabolic bifurcations in complex dimension two
type: Item
instance of: scholarly article
title: Semi-parabolic bifurcations in complex dimension two
zbMATH Open document ID: 1365.37044
author: Eric Bedford
author: John Smillie
author: Tetsuo Ueda
published in: Communications in Mathematical Physics
publication date: 2017-03-01T00:00:00Z
full work available at URL: WRAP-parabolic-bifurcations-complex-Bedford-2017.pdf
review text: Motivated by the study of parabolic bifurcations in one complex variable, the authors study semi-parabolic bifurcations of families of diffeomorphisms in complex dimension two. For the family of maps \(f_\epsilon(z)=z+z^2+\epsilon^2\) in one complex variable, it is known that: 1) the functions \(\epsilon\mapsto J(f_\epsilon)\) and \(\epsilon\mapsto K(f_\epsilon)\) are discontinuous at \(\epsilon=0\), where \(J\) denotes the Julia set and \(K\) denotes the filled Julia set; 2) if \(\epsilon_j\rightarrow 0\) and \(n_j\rightarrow \infty\) are sequences such that \(n_j-\pi/\epsilon_j\rightarrow \alpha\), then \(\lim_{j\rightarrow \infty} f_{\epsilon_j}^{n_j}=t_\alpha\), where \(t_\alpha\) denotes the transition map which maps the attracting petal to the repelling petal. The authors consider the family of maps \(F_\epsilon(x,y)=(x+x^2+\epsilon^2+\mathrm{h.o.t.},b_\epsilon y+\mathrm{h.o.t.})\) with \(|b_\epsilon|<1\). Among other things, they prove results for \(F_\epsilon\) similar to 1) and 2) above. The paper is well-written and contains many interesting figures.
Mathematics Subject Classification ID: 37F45
Mathematics Subject Classification ID: 37F50
Mathematics Subject Classification ID: 32H50
zbMATH DE Number: 6690495
zbMATH Keywords: semi-parabolic bifurcation
zbMATH Keywords: Fatou coordinates
MaRDI profile type: Publication
OpenAlex ID: W2581002007
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DOI: 10.1007/S00220-017-2832-Y
description: scientific article; zbMATH DE number 6690495