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Consider a rational map $f \colon (x_1,\ldots,x_n) \mapsto (P_1(x_1,\ldots,x_n),\ldots,P_n(x_1,\ldots,x_n))$, where the $P_i$ are rational functions. Via iteration this map defines a discrete dynamical system on $\mathbb{k}^n$, where $\mathbb{k}$ is some algebraically closed field (take $\mathbb{k}=\mathbb{C}$ if you want). This system is said to have singularity confinement if all singularities of the system are "confined" in the sense that if there is $\mathbf{x}\in\mathbb{k}^n$ for which $f(\mathbf{x})$ is not defined (because of division-by-zero issues), then there is always some $k\geq 0$ for which $f^k(\mathbf{x})$ is defined.

In fact, for simplicity, in this question we can just consider ``one-dimensional'' dynamics $x_{n+1}=P(x_1,\ldots,x_n)$ for a rational function $P$. So singularity confinement here means that $x_{n+1}$ might not be defined for a certain choice of $x_1,\ldots,x_n\in\mathbb{k}^n$, but then there must be some $k\geq 1$ so that $x_{n+k}$ is defined.

One reason that singularity confinement might hold is because of Laurentification, as explained for instance in Section 4 of https://arxiv.org/abs/1709.00578. This means that there is a sequence of irreducible, coprime Laurent polynomials $y_1,y_2,y_3,\ldots,$ in $x_1,\ldots,x_n$ so that $x_{n+k}=Q(y_{(k-1)\ell+1},\ldots,y_{k\ell})$ for all $k\geq N$ for some fixed rational function $Q$ and $\ell, N\geq 1$.

Is there any example of a rational mapping system which has singularity confinement, but not because of Laurentification? In other words, this would mean that the irreducible factors appearing in $x_{n+k}$ do not eventually form any fixed pattern, but these factors still manage to always "diappear" in a finite amount of time.

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    $\begingroup$ How about $x_3=x_2+\frac{a}{{x_2}^2}-x_1$? It has singularity confinement according to arxiv.org/pdf/solv-int/9711014.pdf. I've coded up for $x_{10}$ using $a=2$ and I don't think it is laurentificable, maybe because it's not positive. $\endgroup$ – Wenze 'Sylvester' Zhang Mar 19 at 23:15
  • $\begingroup$ @Wenze'Sylvester'Zhang: really interesting example! I'll accept it as an answer if you provide more details (about why it has singularity confinement and what the pattern of irreducible factors looks like) $\endgroup$ – Sam Hopkins Mar 19 at 23:24

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