Consider the expression $(1/a)^{\sin(a)+1}$ where $a$ is a real number $0<a$ and not equal to $1$. Sinus in radians. I have iterated with many initial values and have some thoughts about the statistic results. The outcomes seem to be chaotic. I made a few assumptions and have more general thoughts than specific questions.
a. Once a value greater than about $0.2$ and significally lower than $7$ the values stabilizes in this range. What are those values or limits? If a value is close to $1$ it may take some iterations before significally different values are spitted out. As the input approches zero the output get large when it approaches $1$ from below the output approches a from above. How would a probabalistic distribution curve look like? Perhaps it would be similar to normal distribution with values close to $1$ in the middle?
b. I guess you will never get the same value twice. Is it possible to prove this and that the values will always stabilize themselves and that there will never be values out of the range again (if so is the case)?
c. What would happen if we iterated an infinite number of times? I think that the distribution curve would not be valid anymore, since it doesn't matter if the probability for values within a certain range is low as long as it is greater than zero when we deal with infinity. (These properties reminds me of an electron viewed as a field.) The numbers would be infinitely dense within the range, since all numbers have zero length. I also suspect that all values after the first two (initial value and the first outcome) will always be non-algebraic. That means that if we iterate again with a different initial value that is algebraic we will have created a new list with not a single number occuring in both infinitey large lists.
d. However, for each value in the range between $0$ and $1$, there has to be a value higher than 1 that yields the same list. In fact, given that the sin-function is periodic, I strongly suspect there are an inifite number of (likely transcendental) values higher than $1$ that will produce the same list. What are the distance between those numbers for a certain value? (It is, of course, possible to do this starting from any number in any list.)
e. What is the longest sequence we can create before the values stabilize?