# Tagged Questions

**0**

votes

**1**answer

80 views

### Looking for methods/results for explicitly bounding iterations of rational functions

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
$$
...

**7**

votes

**2**answers

604 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 ...

**6**

votes

**4**answers

409 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? ...

**3**

votes

**3**answers

671 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 ...

**7**

votes

**5**answers

1k views

### Rational maps with all critical points fixed

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 ...

**5**

votes

**5**answers

1k views

### When does the sequence of iterates of a rational function converge?

Darsh asks at the 20-questions seminar:
Let f:P^1 -> P^1 be rational function.
Can you say when the sequence ...