Fiber Lyapunov exponent (${\lambda }_{\mathrm{fiber}}$)

A fiber Lyapunov exponent measures infinitesimal growth within the fiber of a differentiable skew product $F(x,y)=(R(x),S_x(y))$, while the base trajectory is held fixed. In a one-dimensional fiber, write $(x_j,y_j)=F^j(x,y)$. When the limit exists,This follows from the chain rule applied to the iterated fiber map. For $S_x(y)=ay+b(x)$ with $a\ne 0$, the exponent is $\log |a|$. A negative fiber exponent describes contraction along that trajectory; it does not by itself assert uniform contraction at every base point.