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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.

A skew product evolves a base coordinate independently while allowing a fiber coordinate to depend on the base:The first coordinate obeys the autonomous rule $x_{n+1}=R(x_n)$, while $y_{n+1}=S_{x_n}(y_n)$ is driven by the base state. Continuity, smoothness or measurability is imposed according to the setting.

For example, $F(\theta ,y)=(\theta +\omega (\mathrm{mod}1),ay+\sin (2\pi \theta ))$ describes a linearly contracting response when $|a|<1$, driven by a circle rotation. Varying $y$ leaves every future base angle unchanged. By contrast, a coupled map whose first coordinate also depends on $y$ is not a skew product in these coordinates.

An invariant graph of a skew product satisfies $h(R(x))=S_x(h(x))$: starting at $(x,h(x))$ lands at $(R(x),h(R(x)))$. The fiber Lyapunov exponent measures response to perturbations within a fiber, separately from growth in the base. This separation is useful for strange nonchaotic attractors.

The irrational skew shift gives an affine example on a torus, while the circle skew-product minimality criterion identifies when a circle extension of a minimal base has no proper closed invariant subsystem.

A strange nonchaotic attractor combines geometrically irregular attraction with no positive Lyapunov exponent for the dynamics under consideration. Systems driven at two incommensurate frequencies supplied the original examples.^[ref]

One useful framework is a skew productThe base is an irrational rotation of the circle. An attracting invariant graph of a skew product obeys $h(\theta +\omega )=F_{\theta }(h(\theta ))$; the geometric question concerns how irregular that graph is. For a differentiable one-dimensional fiber map, the fiber Lyapunov exponent along it is, when the limit exists,A negative value describes average contraction of perturbations within the fiber. The base rotation has a neutral exponent, so a negative fiber exponent does not mean that every exponent of the full forced system is negative. Under appropriate boundedness and integrability assumptions for this triangular derivative system, the full largest exponent is zero rather than positive.

These equations specify a framework, not a claim that every choice of $F_{\theta }$ produces a strange attractor. For instance, $F_{\theta }(x)=ax$ with $|a|<1$ has the smooth attracting graph $h=0$. A negative fiber exponent alone therefore establishes neither geometric strangeness nor a discontinuous graph.