Do two exponential spirals intersect?

Question

I have lists of complex points: orbit of complex point z under quadratic function

 f(z) = z*z

I know that lists are:

• z, z^2, z^4, z^8, ...
• (r,t), (r^2, 2*t), ..., (r^(2^n), t*2^n)

where :

• r = abs(z) is the absolute value
• t = arg(z) is the argument

Let r> 1, then points of the orbit lay on the exponential spirals.

Shape of such spiral depends on the initial point of the orbit: z0

I thought that such spirals do not intersect

Do such curves have common points ? ( intersect)

My naive anlysis shows intersection points

Answer 1

You have $f(x)=x^2$. Let two initial points be $z \neq w \in \mathbb{C}$. The orbits $f^n(z)$ and $f^m(w)$ can intersect if $m \neq n \in \mathbb{N}$. Suppose $m = 1$ and $n = 2$, then the orbits intersect when $$w^2 = z^4$$ which has the solutions $$z = \pm \sqrt w \\ z = \pm i \sqrt w$$