# Questions tagged [algebraic-dynamics]

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8
questions

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### Is the Lyness 5-cycle map birationally conjugate to its own square?

Let $L(x,y) = (y,(y+1)/x)$. On a dense open subset of the plane, $L$ and all its powers are well-defined invertible maps and $L^5$ is the identity ($L$ is sometimes called the Lyness 5-cycle map). $L$ ...

3
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0
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### Irregularly Intertwined Linear Recursions: Other References?

I was wondering if anyone had run across the following notion of intertwined linear recursions. I'm looking for references, or even a standard name. (I know one source, which is the genesis of this ...

4
votes

0
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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 $\mathbb{Q}_p$-...

3
votes

1
answer

286
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### 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?

2
votes

0
answers

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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 ...

3
votes

0
answers

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### 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 $\mathcal{O}_X$-...

4
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1
answer

162
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### 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 ...

12
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### 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 ...