fleet charm
I'm studying the behaviors of a family of functions f(x) = (x - x mod b)/b + k mod b parametrized by integers b, k, as a dynamic system. I only have around 2 weeks left and I was always stuck having to describe anything about the periodic points, coz as u can see, iterates of f isnt really easy to formulate. Then, today I just realized that the graph of f(x) looks just the same as a Bernoulli map B(x) = kx mod 1, which is much more studied and hopefully can give me a direction so I could finally finish my paper aaa. Anyway, yes, f(x) looks like a Bernoulli map with parameter k, but the graph is linearly transformed with the following matrix:
bk 0
k bk
But it's the graph that is transformed, and I think this is more like a 2d thing? and I'm trying to study f(x), a 1D map. So I'm not much sure how precisely it should be stated and here I am asking for help. For now, what I know is that the line y = x, with which the cobweb diagrams are used, will be transformed as well. I have already calculated it'll turn to y = (b - 1)x/b, so the corresponding Bernoulli map variation will be x_(n+1) = [(b - 1)/b] * (kx mod 1)
Also, this is for a research paper, I'm just a senior high student and I didn't really have experience writing math papers so I would also like at least your opinion on what exactly should be a proposition, a theorem, and maybe a property (in a paper). What I found out so far is it's mostly subjective, a theorem is much more important proposition, a property (statement) I just found in a paper related to whatever I'm working rn.