For a personal project of mine, I need a mathematical function which generates values. In Computer Science terms, this is a PRNG, however I can't find one applicable to my problem. $h(x_0, params) = x_1$ for most values, but sometimes $h(x, params) = x$, and this should be dependent on the params.
I am not a mathematician, so I can't really think of, or find, a function that these conditions apply to.
For clarification, essentially, I would like to find a pseudorandom number generator that gets stuck on a number, and will get stuck on that number after any number of iterations.
For even greater clarification, the value upon which the PRNG gets stuck on would be determined by the params passed into the function as well, and this number or set of numbers would remain constant, no matter the previous value.
I've considered elliptic curves, however the $h(x) = x$ does not hold true for them.
Thank you for any answers, and feel free to tell me how to word it better.