07/08/2022, 03:16 AM
Also, to Catullus' question, there is a pattern to the fixed points.
In Devaney's book "An intro to chaotic dynamical systems" He has a large section on iterating:
\[
\lambda e^z\,\,\text{for}\,\,0 < \lambda\\
\]
He describes the structure of the Julia set/fatou sets using symbolic dynamics. It's far too involved to describe here, but it's a standard modeling technique in dynamics. Devaney describes how different parts of the complex plane maps to other parts under iterations, and on intersections of these domains we have the fixedpoints/periodic points. It's absolutely fascinating. I'm not that well versed in it, but to start you off, search for the symbolic dynamics of the exponential functions. Or pick up Devaney's book
In Devaney's book "An intro to chaotic dynamical systems" He has a large section on iterating:
\[
\lambda e^z\,\,\text{for}\,\,0 < \lambda\\
\]
He describes the structure of the Julia set/fatou sets using symbolic dynamics. It's far too involved to describe here, but it's a standard modeling technique in dynamics. Devaney describes how different parts of the complex plane maps to other parts under iterations, and on intersections of these domains we have the fixedpoints/periodic points. It's absolutely fascinating. I'm not that well versed in it, but to start you off, search for the symbolic dynamics of the exponential functions. Or pick up Devaney's book
