Semi-parabolic bifurcations in complex dimension two
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.
scientific article; zbMATH DE number 6690495