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### Finiteness of the set of $\mathbb{Q}_p$-rational periodic points

The statement I am concerned with is this:
Let $\varphi : \mathbb{P}^r_{\mathbb{Z}_p} \to \mathbb{P}^r_{\mathbb{Z}_p}$ be a morphism of degree higher than one. Then the set of ...

**3**

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201 views

### Hypersurfaces with rational self-maps

I'm looking for interesting examples of hypersurfaces $X\subset \mathbb P^n$ with a rational self-map $X\dashrightarrow X$?
Are there such examples for cubic hypersurfaces?

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### Must the coordinates of a polynomial iteration have about the same size?

Original post. The following statements seem plausible (not to say intuitively obvious), but I do not see how to prove them.
Let us say that a polynomial mapping of $\mathbb{C}^2$ is reducible if ...

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71 views

### Question about a length inequality in algebraic dynamics

Let $X$ be a Noetherian scheme. Let $f\colon X\rightarrow X$ be an integral self-morphism. If $x\in X$ is a closed point, I will write $\mathcal{F}_{1}^x$ for the coherent sheaf of ...

**4**

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127 views

### If the sequence of degrees of the iterates of a self-map of $\mathbb{A}^2$ is bounded, is it eventually periodic?

Let $f : \mathbb{A}_k^2 \to \mathbb{A}_k^2$ be a regular self-map of the affine plane over a field $k$ of characteristic zero. Assume that the sequence $(\deg{f^n})_{n \in \mathbb{N}}$ is bounded. Is ...

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586 views

### Rational iterations on $\mathbb{P}^1$ defined over $\mathbb{Q}$ and possessing a totally $2$-adic point of a high finite order

A (final) remark (9/29). Now that the question is open for bounty anyway, and hence cannot be deleted even though everything boils down to the line added on 9/28, I may as well record -- for anyone ...