# A different type of infinite power tower function

The question is about the following: Let a function be defined such that $$f_n(x) = x \uparrow \uparrow n$$ Where $$n$$ is a natural number

Now, it is reported at many places that the function $$F_1(x) = \lim_{n \rightarrow \infty} f_n(x)$$ Is defined only for $$x \in [e^{-e} ,e ^{\frac {1}{e}}]$$

For bigger $$x$$, the limit doesn't exist because the 'power tower' is not 'convergent' , it just keeps getting bigger.

For small $$x$$ , however, this is not the case. I tried to approximate this function for $$x = 0.05$$ on a calculator and did some 20-30 iterations. And the value seemed to oscillate between $$0.137$$ and $$0.663$$.

(Which is strange because $$0.05^x = x$$ has a solution approximately at $$0.3502$$)

So, I have two questions:

$$(1)$$ Is it true that the 'power tower' oscillates between two distinct limiting values or is it converging to a single value very slowly?

$$(2)$$ If $$y = F_1(x)$$ then for $$x \in [e^{-e} , e^{\frac {1}{e}}]$$ We can write $$y=x^y$$ So, for $$x < e^{-e}$$ , what implicit/explicit relations exist for the following functions: $$F_2(x) = \lim_{n \rightarrow \infty} f_{2n+1} (x)$$ And $$F_3(x) = \lim_{n \rightarrow \infty} f_{2n} (x)$$

Also, it would be appreciated if anyone can provide some intuition (preferably without involving the Lambert function) about why the lower bound of $$e^{-e}$$ exists.

(For example, the 'intuition' for the upper bound of $$e^{\frac {1}{e}}$$ comes from the fact that this is (almost) $$x=y^{\frac {1}{y}}$$ and $$y^{\frac {1}{y}}$$ attains a maximum value of $$e^{\frac {1}{e}}$$ which is evident from equating the derivative to zero)

• It somewhat oscillates and does not converge, even if it is bounded. See the image on the wikipedia entry. en.wikipedia.org/wiki/Tetration
– SK19
Apr 13, 2019 at 11:18
• So the values it oscillates between.... What is the relation (implicit/explicit) between them and the input $x$? Apr 13, 2019 at 11:22
• This sort of behavior in iterated functions has been studied for 40+ years now, and led to chaos theory. It has been too long since I last tooked into it for me to be able to say much, but you can see similar behavior with iterating the function $f(x) = \lambda4x(1-x)$ on $[0,1]$. For low $\lambda$, sequences not starting at $x = 0$ will converge to the root of $f(x) = x$. But for high enough $\lambda$ it splits to alternating between the two roots of $f(f(x)) = x$, for higher yet, it goes to the 4 roots of $f(f(f(f(x)))) = 0$, then to 8 roots, etc, until it finally turns chaotic. Apr 13, 2019 at 23:01
• @PaulSinclair Thanks a lot for the reply. I wanted to ask one more thing, in reference to the question in the original post, I guessed that $F_2(x)$ and $F_3(x)$ should satisfy $y = x^{x^{y}}$ instead of $y = x^{y}$ and this seems to work but I don't know what is the right way to deduce that. Also, for even smaller $x$, does the infinite power tower (as described above) split into alternating between $3$ or $4$ values? . Also, I couldn't find any sources on the internet that discuss the functions $F_2(x)$ and $F_3(x)$, can you please point me to some that you happen to know about? Apr 23, 2019 at 12:53
• I spent a little time looking at this a few decades ago. I don't qualify as an expert. However, I did some searching and discovered it is called Bifurcation theory. If I recall correctly, for a one-dimensional recursion, it bifurcates when the slope of the function at the fixed point $x = f(x,\lambda)$ drops below $-1$. Apr 23, 2019 at 23:41

I would like to point out that I still have one unanswered question that is not addressed in this paper. We know that for $$x \in [e^{-e} , e^{1/e}]$$ $$F_1(x) = \frac {W (- \ln x)}{- \ln x}$$ Where $$W$$ is the Lambert function
Similar to this, what is the explicit expression for $$F_2(x), F_3(x)$$ ? The paper even provides an infinite sum for $$F_1(x)$$ $$F_1(x) = \sum_{n=0}^{\infty} \frac {(n+1)^n (\ln x)^n}{(n+1)!}$$ Can some infinite sum be generated for $$F_2(x) , F_3(x)$$ also?