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.