Probability to get into the segment of [1,5000] after iterations with the logarithm

Question

Let $$g(x) = \begin{cases} x, if \quad x \in [1,10000] \\ \lg(\lg(\ldots \lg(x))), \text{ Iterated until the result is less than 10,000} \end{cases} $$ and $g : \mathbb{N} \to \mathbb{R} $, What is the probability that the number will be less than $5000$? If probability is defined as $S_n :=$ the number of numbers for which $g(x)$ is less than $5000$, then the probability is $$ \lim_{n \to \infty} \frac{S_n}{n} $$

Answer 1

The limit doesn't exist. At $10^{10^{5000}}$, the ratio is almost exactly $1$, and at $10^{10^{10000}}$ it's almost exactly $0$, and it will continue to oscillate like that because the stretches grow so quickly and make everything that happened before irrelevant.