Question: Let $f(x)$ be a polynomial in $\mathbb{Z[x]}$. Is there a relation between the property $P_i$ of $f$ and the number of its periodic points with period $p$ (x is a $p$-periodic point of $f$ if $\exists p:f^p(x)=(f\circ f\circ\cdots\circ f)(x)=x$; and it is a periodic point with period $p$ if it is $p$-periodic but not $q$-periodic $\forall q< p$)?

$P_1: \deg (f) \sim n$

$P_2: sign(\deg(f)) \sim parity(n)$

$P_3:$ number of (possibly real) roots of $f$

$P_4: D(f)$ (the discriminant of $f$)

$P_5: |f'|\leq M_1, |f''|\leq M_2, |f'''|\leq M_3,...$ (if some derivative of $f$ bounded)

  • We can play with the domain of $f$ and the set its coefficients are from (for instance what if $f\in\mathbb{Q}[x]$ or $f\in\mathbb{F}_p[x]$, though the latter one may have no significant applications).

Motivation: I am studying Barreira & Valls' Dynamical Systems: An Introduction as a reading course, and I came upon two questions asking for periodic points of specific polynomials (which happen to be the first two exercises in the book, on p. 24). The first one asks for the periodic points with period 2 of $f:\mathbb{R}\to\mathbb{R},x\mapsto 3x-3x^2$ and the second one asks for the periodic points with period 5 of $f:\mathbb{R}\to\mathbb{R},x\mapsto x^2+1$. The first one has 2-periodic points, but none of them have period 2 (one is 0 and the other is 2/3, which is a fixed point). I am still thinking about the second one, but it might be useful that at each iteration the function shifts upward along the $y$-axis ($f\geq1,f^2\geq2,f^3\geq5,f^4\geq26,...$) and the iteration is decreasing when $x<0$ and increasing when $x>0$ (by taking the derivative, which is seen to be a polynomial with terms only in odd degree). I thought that it would be convenient if we had a general result about the periodic points of a polynomial, hence my question.

Disclaimer: The properties that I listed are the first ones that came to my mind, and may be absolutely irrelevant. Also I remembered to google "periodic points of polynomials" just before I was about to post the question, and there are a few resources (which I have not checked yet). But I don't think there is a similar question on SE, so my question may be helpful as future reference.


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