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In Theorem 2.6.4 of Beardon's book, "Iteration of Rational Functions", he states the values for the first two coefficients of an iterated power series. That is, suppose that $$... 2answers 657 views ### Is this a Julia set (and if so, for which function family is it the Julia set)? Consider the function family given by f_\lambda(z) = z - p_\lambda(z)/p_\lambda'(z) where p_\lambda(z) = (z^2 - 1)(z - \lambda). Every attracting cycle and every rational neutral cycle of ... 4answers 441 views ### What are good methods for detecting parabolic components and Siegel disk components in the Fatou set of a rational function? Given a rational function f on the Riemann sphere, I would like to answer the question: Does the Fatou set of the function, F(f), contain any parabolic components or Siegel disk components? ... 3answers 688 views ### When a sequence of coefficients converges to the coefficients of a rational function R, when does the sequence R_n converge uniformly to R? Let R be a rational function of degree d mapping the Riemann sphere to itself:$$R(z) = \frac{a_d z^d + a_{d-1} z^{d-1} + \cdots + a_0}{b_d z^d + b_{d-1} z^{d-1} + \cdots + b_0} where $a_d$ and ...
What can be said about rational self-maps of $\mathbb P^1$ for which all critical points are also fixed points ? If all but one of the fixed points are critical, there is a characterization in ...