I would like to know if there are cycles in some specific functions other than the Collatz function. Let's consider function:
$f(x) = \frac{a}{2} \cdot x + \frac{b}{2}$ if x is odd
$f(x) = \frac{x}{2}$ if x is even
Of course it is easy to see that for example $a=7$ and $b=5$ got cycles. But I'm searching for such $a$ and $b$ that there are no cycles. And it will be best if $a$ and $b$ will be big numbers - about 64-128-bit long. Let's say I will choose such random numbers. What's more let's say it is no problem if cycle is longer than $128 \cdot 2^{128}$ and/or if the samllest number in cycle will be bigger than $2^{128}$ - it is also fine. Is there any chance to find such funtions (without cycles or only with longer/bigger cycles) according to current knowledge? I was trying to extend somehow proof by Simons and de Weger:
https://deweger.xs4all.nl/papers/[35]SidW-3n+1-ActaArith[2005].pdf
but I don't understand much of this work. We can't check this directly, computationally, because the numbers are too big.