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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)

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  • $\begingroup$ It somewhat oscillates and does not converge, even if it is bounded. See the image on the wikipedia entry. en.wikipedia.org/wiki/Tetration $\endgroup$
    – SK19
    Apr 13, 2019 at 11:18
  • $\begingroup$ So the values it oscillates between.... What is the relation (implicit/explicit) between them and the input $x$? $\endgroup$
    – anonymous
    Apr 13, 2019 at 11:22
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    $\begingroup$ 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. $\endgroup$ Apr 13, 2019 at 23:01
  • $\begingroup$ @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? $\endgroup$
    – anonymous
    Apr 23, 2019 at 12:53
  • $\begingroup$ 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 $-1$. $\endgroup$ Apr 23, 2019 at 23:41

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While searching online for relevant articles on infinite exponentiation I came across this one by R Arthur Knoebel

http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3087&pf=1

It seems to answer almost all my questions so I decided to put it here as an answer for anyone else to see too.

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?

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