Dynamics of $e^z+z^2+z+1$

Question

Question: How to find the dynamics of $f(z)=e^z+z^2+z+1$ also how to find the escaping set ?

Since critical points control the dynamics, I want to find the critical points first, but I am not getting any explicit expression for the critical points also I didn't find any literature regarding this. I have shown that along any unbounded curve, the function remains unbounded. Will this idea help me anyway?

Can someone guide me on how to handle this kind of situation? Thank you.

Answer 1

According to the picture below, it seems the Julia set is the whole plane, and escaping set is dense. Darker parts escape faster (and possibly come back by the effect of $e^z$).

$f(z)$ in $[-2\pi,2\pi]^2$ " />

1. In the right half plane (i.e., when $\mathop{\rm Re}(z)$ is large), the dynamics is close to $e^z$, so there are Cantor sets of hairs like the result of Devaney-Krych.
2. Outside the half plane and when $|z|$ is large, the dynamics is close to $z^2$, so eventually almost every point eventually enter the right half plane.
3. I haven't checked precisely, but from the picture it looks like that there is a quadratic-like restriction near the origin, whose Julia set is a Cantor set and some of the hairs land at it.

Of course we need much more precise estimate, but probably those three parts dominate the dynamics.