Invariant graph of a skew product ($h(R(x))=S_x(h(x))$)

An invariant graph of a skew product is a function $h$ whose graph is preserved by $F(x,y)=(R(x),S_x(y))$. Its defining relation is $h(Rx)=S_x(h(x))$: the image of $(x,h(x))$ lies over the new base point on the same graph.

For an invertible base and $S_x(y)=ay+b(x)$ with $|a|<1$ and bounded $b$, an explicit bounded solution isAbsolute convergence follows from a geometric bound, and shifting the summation verifies the invariance equation. The formula shows how the current response combines the entire past forcing, with older contributions suppressed by powers of the contraction factor.