What is difference between Convergence and Fixed point?

Question

i have recently studied fixed points but it left some questions to me.

i saw a function $x^{x^{x^{.^{.^{.}}}}}$, leading to $x^y=y$ when it converges to the value $y$.

for instance, we can find fixed points $y=2$ and $y=4$ when $x=\sqrt{2}$. However it only converges to $y=2$ despite we supposed like "when it converges to the value $y$"

Hence eventually, it seems like it is leading to a fixed points not a point which is converging. it likely means we should check if it converges by using any graphical tools etc.

but my question is, why people say "when it converges to the value y" not "we have to find fixed points so we will use equation like $x^y=y$"?

Answer 1

If the infinite power tower $x\uparrow x\uparrow x\uparrow \cdots$ converges to some limit $L$, $L$ must satisfy $x^L=L$ , so it must be one of the solutions of $x^L=L$. But $x^L=L$ does not mean that the power tower converges to $L$ , not even that it converges.