Continuity of Julia sets and its Hausdorff dimension of \(P_{c}(z) = z^{d} + c\)
Let \(R\) be a rational map of degree \(d = \deg{R} \geq 2\) on the complex sphere \(\widehat{\mathbb{C}}\), and denote by \(R^n\) the \(n\)-th iterate of \(R\). The Fatou set \(F(R)\) is the maximal open set in which the sequence \((R^n)\) is a normal family in the sense of Montel, while the complement of \(F(R)\) in \(\widehat{\mathbb{C}}\) is the Julia set \(J(R)\). It is well known that \(J(R)\) is a perfect set, and thus uncountable. If it is disconnected, then it has infinitely many components.NEWLINENEWLINEIf \(\mathcal{C}\) denotes the set of critical points of \(R\) then the set of critical values of \(R^n\) is given by NEWLINE\[NEWLINE Ctv_n(R) = R(\mathcal{C}) \cup R^2(\mathcal{C}) \cup\dotsb\cup R^n(\mathcal{C}). NEWLINE\]NEWLINE Then the \(\omega\)-limit set \(\Omega(R)\) is the set of all \(z \in \widehat{\mathbb{C}}\) such that there exists \(c \in Ctv_n(R)\) and a sequence \((n_k)\) of positive integers such that \(n_k \to \infty\) as \(k\to\infty\) and \(R^{n_k}(c) \to z\) as \(k\to\infty\).NEWLINENEWLINEThe author calls a critical point \(c\) of \(R\) recurrent if \(c \in \Omega(R)\) and non-recurrent otherwise. Such maps are denoted as NCP maps. They are called semi-hyperbolic if the critical points are not recurrent by \(R\) and without parabolic cycles.NEWLINENEWLINEA sequence \((R_n)\) of rational maps converges to \(R\) \textit{algebraically} if \(\deg{R_n}=\deg{R}\) and, when \(R_n\) is expressed as the quotient of two polynomials, the coefficients of which can be chosen such that they converge to those of \(R\). Equivalently, there holds \(R_n \to R\) as \(n\to\infty\) in the spherical metric.NEWLINENEWLINENow, let \(R_n \to R\) as \(n\to\infty\) algebraically, and let \(b \in J(R)\) be a pre-periodic critical point with \(R^j(b)=R^k(b)\) for some \(j>k>0\). Furthermore, suppose that for all such \(b\) and for all sufficiently large \(n\) the maps \(R_n\) have critical points \(b_n \in J(R_n)\) with the same multiplicity as \(b\), \(b_n \to b\) as \(n\to\infty\) and \(R^j(b_n)=R^k(b_n)\). Then the author says that \(R_n \to R\) as \(n\to\infty\) \textit{preserving critical relations}.NEWLINENEWLINEFinally, let \(HD\) denote the Hausdorff dimension and \(d_H\) the Hausdorff distance.NEWLINENEWLINEIn this paper the author studies the dynamics of polynomials of the form \(P_c(z)=z^d+c\), \(d \geq 2\) such that the critical point \(0\) is not recurrent and \(0 \in J(P_c)\). These polynomials are semi-hyperbolic in the sense of \textit{L.~Carleson} et al. [Bol. Soc. Bras. Mat., Nova Sér. 25, No. 1, 1--30 (1994; Zbl 0804.30023)]. Furthermore, let \(\mathcal{M}_d\) be the set of all \(c\) such that \(J(P_c)\) is connected. It is called the connectedness locus, and for \(d=2\) it is the famous Mandelbrot set. Now, the main theorem of the paper reads as follows.NEWLINENEWLINELet \(c_0 \in \partial\mathcal{M}_d\) such that \(P_{c_0}\) is semi-hyperbolic. If \(P_{c_n} \to P_{c_0}\) as \(n\to\infty\) algebraically, then there is some positive constant \(C\) with NEWLINE\[NEWLINE d_H(J(P_{c_n}),J(P_{c_0})) \leq C|c_n-c_0|^{1/d}\,. NEWLINE\]NEWLINE If, in addition, \(P_{c_n} \to P_{c_0}\) as \(n\to\infty\) preserving critical relations, then \(P_{c_n}\) is semi-hyperbolic for all sufficiently large \(n\), and \(HD(J(P_{c_n})) \to HD(J(P_{c_0}))\) as \(n\to\infty\).NEWLINENEWLINETools of the proof are holomorphic motions, a characterization of stability, conformal measures, conical sets, the construction of a net and a bounded distortion property (which is a consequence of the Koebe distortion theorem).
scientific article; zbMATH DE number 6232361