real/integer quadratic mapping's real/integer k-periodic points

Question

The quadratic mapping is discrete dynamical system $$x_n = x_{n-1}^2 + c$$ There is systematic reference about periodic points when $x_n$ and $c$ are both complex. $k$-periodic points are $x_{n+k} = x_{n}$ for some finite $k$.

However, I'm curious about periodic points of this dynamic system when both $x_n$ and $c$ are real or integer. First of all, when $c >= 1$, there is no real periodic point.

My question:

1. Are there systematic references for real/integer quadratic map?

2. I'm curious about what's the existence of real $k$-periodic points for other real $c$ and existence of integer periodic points for integer $c$. If exist, how to find all periodic points for both real and integer case?

Answer 1

The iteration of real functions in general and real quadratic functions in particular has been well studied and there are a ton of elementary introductions at this point. I think a reasonable introduction can be found here. In particular, the last exercise in the section on conjugacy asks you to

Show that $f(x)=x^2+(2\lambda-\lambda^2)/4$ is conjugate to $g(x)=\lambda x(1-x)$ via the conjugacy $\varphi(x) = -\lambda x + \lambda/2\text{.}$

That answers your question in the comments about the equivalence between the quadratic map and the logistic map.

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Much less has been written about iteration on the integers. Perhaps, that's because it's very hard; the theorems of calculus that we have in complex and real iteration simply don't apply when we restrict the function to the integers. More likely, though, periodic orbits under the iteration of polynomials are unlikely to live within the integers, since the solutions of high order polynomials are unlikely to be integers. Nonetheless, here are a couple of conjectures to whet your appetite:

1. Every odd integer is a fixed point of $f_c$ for some integer $c<0$.
2. Every even integer is a point of period two under iteration of $f_c$ for some integer $c<0$.

I don't believe that there are any integer orbits for integer values of $c$ when the period of the orbit is larger than 2. I don't know that for sure, though.