Depending on $\alpha,N,x$ the sequence $x\mapsto x^\alpha \mod N$ can have a different length. If the first element $x_0$ is initialized with $x_0 = x_r^\alpha$ for a random $x_r$ the sequence will almost always have the same constant size.
We will focus here only at the most common sequences with max size $N_L$ (for given $\alpha,N$).
Depending on chosen $x_0$ it can lead to different, disjoint sequences with still the same max sequence size.
Question: Is there some general formula to compute the number of those sequences (for given $\alpha,N$)?
Edit: The posted & accepted answer did not answered this question but was very helpful.
An answer to this question is still welcome. (edit end)
While tinkering around I already found some formula for some $N, \alpha$ of special structure. The cycle length $\alpha$ modulo some prime factors of $\phi(N)$ and also the factors of $\phi(\phi(N))$ seem to have some impact on that count.
It's also related to the number of different values $N_{\alpha}=|\{x^\alpha \bmod N\}|$ and the max length of those sequence $N_L$.
For $N_\alpha$ it got some answer in another thread. If $N$ is a product of unique prime factors:
$$N = \prod_{i=1}^n p_i$$
The number of different values $N_\alpha$ would be
$$N_{\alpha} =\prod_{i=1}^n\left(1+\frac{p_i-1}{\mathrm{gcd}(\alpha,p_i-1)}\right)$$
For $N_L$ I only made up some equations which do work out for some $N, \alpha$. A general formula would also be welcome.
Both together could lead to an approximation of the sequence count.
Side-question: Do those sequence have a special name? Is this and other properties described somewhere (in a compact form)?
The target $N$ will also have $2$,$3$ or $4$ odd unique prime factors.
