= Fiber Lyapunov exponent
{title2=$\lambda_{\mathrm{fiber}}$}
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,
$$
\lambda_{\mathrm f}(x,y)=\lim_{n\to\infty}\frac1n\sum_{j=0}^{n-1}\log|\partial_yS_{x_j}(y_j)|.
$$
This follows from the <chain rule> applied to the iterated fiber map. For $S_x(y)=ay+b(x)$ with $a\ne0$, 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.
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