Forward-invariant set ($F(S)\subseteq S$)

A subset of a dynamical system is forward invariant if every forward iterate, or every forward-time trajectory beginning in the set, stays in it. For a map $F$, this means $F(S)\subseteq S$. It does not require backward invariance or equality $F(S)=S$. Such sets give useful positivity and boundedness guarantees.

An invariant interval for a scalar map is an interval whose points remain in that interval under every forward iterate. Here invariance means forward invariance and does not assert surjectivity onto the interval.

For $F(N)=(N-1/2)(N-1)(2-N)+1/2$, every positive initial value with $F(N_0)>0$ has positive subsequent iterates. Indeed $F(N)<3/2$ for every $N>0$, since $F(N)-3/2=-N(2N^2-7N+7)/2<0$. On $[0,3/2]$, the negative part of the cubic product has magnitude at most $3/8$, so $F(N)\ge 1/8$. Thus the first positive image enters the forward-invariant interval $(0,3/2)$.