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I'm really into weird equations, and in my boredom I came up with this monstrosity.

Let $$ Z(t) = 1 + 1/t^{1 + 1/(t + 1)^{1 + 1/(t + 2)^{1 + 1/(t + 3)^{\dots}}}} $$ and define $f^{\circ\ n}(x) = (f \circ f \circ f \dots \circ f)(x)$ where there are $n$ compositions of $f$.

By looking at graphs of $Z^{\circ\ n}$ for $n = 1, 10, 100, 1000$, I conjecture that $Z^{\circ\ \infty} \equiv\lim_{n\to\infty}Z^{\circ\ n}(x)$ is a constant function. How can this be proven or disproven? If it is a constant function, what is the value of $Z^{\circ\ \infty}$?

user3002473
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1 Answers1

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One observation, albeit trivial, is that $$Z(t)=1+t^{-Z(t+1)}$$ One line of thought is to try to show using this structure of $Z(t)$ that it is a contraction mapping on some interval, assumedly $[1,\infty]$. Then, by Banach's fixed point theorem, the above mentioned sequence of iterated mappings is going to converge to a unique point $t_0$ where $t_0$ is the solution to $Z(t)=t$. The hard part is to prove that $Z(\cdot)$ is a contraction mapping, if at all.

Samrat Mukhopadhyay
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