On explicit connections between dynamical and parameter spaces
Let \(M_\ell\) denote the Mandelbrot set for the family of polynomials \(f_c: S^2\to S^2\) of the form \(f_c(z)= z^\ell+ c\) of the Riemann sphere and \(K_c\) the filled in Julia set of \(f_c\), where \(\ell\) is fixed. It is shown that there exists a constant \(L_*\) such that \[ \text{dist}(c,\partial M_\ell)\leq L_*\ell^2\int_{K_c} |z|^{l-2} dz, \] where \(dz\) denotes the Lebesgue measure on the plane. This result is then refined for hyperbolic and nonhyperbolic components of \(M_\ell\). The second result of the paper deals with the velocity of motion of periodic orbits. Let \(b_1,\dots, b_n\) be an orbit of \(f_c\) of prime period and with multiplier \(\rho\not\in\{0,1\}\). Since \(\rho\neq 1\) there is a holomorphic extension of the orbit to a periodic orbit (of the same period) of \(f_{\widehat c}\) in some neighbourhood of \(c\) with multiplier \(\rho(\widehat c)\), which is a holomorphic function. Defining \[ A(z)= \sum^n_{k=1} {1\over (z- b_k)^2}+ {(f^n_c)''(b_k)\over \rho(1- \rho)(z- b_k)}, \] the relation between the parameter plane (represented by the derivative \(\rho'\)) and the dynamical plane is given by \[ A(z)= (TA)(z)+ {\rho'(c)\over \rho(c)(z- c)}. \] Here, \(T\) denotes the transfer operator. Many applications and corollaries are given (including the Douady-Hubbard-Sullivan theorem). The paper concludes with an extension of the results to polynomial-like mappings.
scientific article; zbMATH DE number 2103499
scientific article