What's the difference between a total basin of attraction and an immediate basin of attraction?

Question

I understand that for an attracting fixed point $\hat{p}$ of a holomorphic self-map defined on some Riemann surface $S$ we define the total basin of attraction as $\mathcal{A}=\text{Bas}(\hat{p})=\{p\in S: (f^{\circ n}(p))_{n\geq 1}\to\hat{p}\}$. Now, in his book, Dynamics of One Complex Variable (Pg. 79 - 3rd-edition), Milnor defines what's called an immediate basin of attraction $\mathcal{A}_0$ to be the connected component of $\mathcal{A}$ which contains $\hat{p}$. I don't understand how this is any different from the total basin of attraction of $\hat{p}$. Can someone provide an example that can clears the difference between these two ideas?

I am attaching a plot of the filled Julia set of $f(z)=z^5+(0.8+0.8i)z^4+z$ which has the following fixed points

$\hat{p_1} = 0$ with multiplier $|\lambda_1| = |f'(0)| =1$ (parabolic),

$\hat{p_2} = -0.8-0.8i$ with multiplier $|\lambda_2| = |f'(-0.8-0.8i)| \approx |-13.7| > 1$ (repelling),

$\hat{p_3} = \infty$ with multiplier $|\lambda_3| = |\lim_{z\to\infty} f(z)|^{-1} = 0$ (super-attracting)

Since the only attracting fixed point here is $\infty$, $\text{Bas}(\infty)=\mathcal{A}=\{p\in\hat{\mathbb{C}}: (f^{\circ n}(z))_{n\geq 1}\to\infty\}$. But what is $\mathcal{A}_0$ in this case?

Answer 1

In general immediate basin od attraction consist of components containing periodic points (= periodic components ) without preperiodic components. Total basin contains both types. If basin consist of only one component then both types are equal.

In your case

• $\hat{p_3} = \infty$ : for polynomials total basin of attraction of infinity is a one component and is equal to immediate basin. Here it is a black component
• $\hat{p_1} $ parabolic point has also i'ts own basin of attraction. The differences are: the speed of attraction is very low and periodic ( parabolic) point is not inside component but on the boundary. One have to use traps ( petals ) to show such dynamics. The immediate basin of attraction for parabolic point will be the union of components for which the parabolic point is a common point ( root point), here = 3 biggest white componnets. Total basin is the union of white components
• $\hat{p_2}$ is attracting point because $|f'(\hat{p_2})| = 0.63840000000 < 1$ and it's immediate basin of attraction is biggest red componnet. Total basin is the union of red components

It is easier to see it for map with single critical point and single periodic orbit, like quadratic polynomial.

Here is parabolic case with repelling petals ( white) and attracting petal ( black triangle)

In superattractive case, like Douady rabbit, immediate basin consist of 3 components ( darker with periodic points inside). Total basin is a whole interior.

f(z) = z^2 is the easiest case. Here the plane ( complexe sphere) consist of 2 components :

• one component = interior : the total and immediate basins are equal ( green color)
• second component is the exterior : the total and immediate basins are equal ( magenta)