Are there any hidden relationships between two numbers $a$ and $b$ such that $a^a=b^b$ when $a \neq b$?
I am attempting to do some research into this relationship, and I came across this when looking at the graph $f(x)=x^x$, and saw that this function has a decreasing interval, before it increases again. I was then intrigued by the idea that this function decreases in the first place, and that there are numbers that can be described by $a^a=b^b$ such that $a \neq b$.
Somethings I've tried. I first looked at how this function behaves when graphed in the imaginary plane, taking ideas from the following paper: https://www.jstor.org/stable/2691469?seq=1. I wasn't really able to find anything special here, although some parts of the paper are above my current level of knowledge. Essentially it looks like the author was able to rewrite $x^x$ (through a process that would take an extra paragraph or two to explain) into this equation $$|x|^x(\cos(nπx)+i\sin(nπx))$$ where n is even for $x>0$ and $n$ is odd for $x<0$. Comparing this to $f(x)=x^x$ I didn't find any action around my original search for numbers $a^a=b^b$.
I also set up this relationship into a relation and graphed it. Graphing $y^y=x^x$ gives a graph that looks like $y=x$ for $x>0$, but it also has interesting behavior around $[0, 1]$. It creates a curve that represents all of the numbers which I am looking for. Graphing this quickly in Desmos gives a little better description, but I am not allowed to imbed pictures yet so I cannot show this.
I also tried expanding through a Taylor Series and looking at the numbers in a table directly. Nothing exciting showed up which I could see.
For context, I am a senior in highschool, and so my knowledge of math is $\leq$ Calc 2. I know very basic set notation if you want to use that in your response.
TL;DR If anyone happens to know anything intresting about $x^x$ or numbers described by $a^a=b^b$ when $a\neq b$ I would love to hear what you have to say.
