Eigenvector of a matrix in which values normalise after multiplication

Question

Definition: Suppose $x\in\mathbb R$. Then define $||x||$ the collapsing of $x$ by $||x||=0$ if $x<1$, and $||x||=1$ if $x\geq 1$.

Definition: Suppose that

• $M$ is a matrix over the closed interval $[0,1]$ (so an entry in $M$ is some fraction of $1$), and
• $V$ is a vector of values from $\{0,1\}$ (so an entry in $V$ is either $0$ or $1$, but nothing in-between).

Then:

1. Write $M_i V$ (as usual) for the sum $\sum_j M_{ij} V_j$. Note that this sum need not necessarily be in $[0,1]$.
2. Define a collapsing application $M\cdot V$ of $M$ to $V$ to be a vector such that the $i$th value in $M\cdot V$ is $||M_i V||$. Note that this is a vector with elements in $\{0,1\}$, like $V$.
3. Call $V$ a collapsed eigenvector of $M$ when $M\cdot V=V$.

Remark: Intuitively, think of $V$ as a list of Boolean inputs, and think of $M$ as a way of combining them, and think of $1$ as a threshold that needs to be passed for this combination to be valid; so the $i$th entry $(M\cdot V)_i$ is $0$ if the combination $M_i V$ is not large enough (collapses to $0$), and $(M\cdot V)_i$ is $1$ if it is large enough (so collapses to $1$).

This is a natural situation where we have a process that proceeds until some deadline at which point it either succeeds or fails. E.g. a voting system, where candidates must exceed a certain threshold of votes to pass to the next round.

Question: Has anyone seen something like this before, and if so, could they provide references --- especially for the theory of collapsed eigenvectors?

Thank you.