= Expanding interval map
{title2=$\inf|T'|=\lambda>1$}
A piecewise differentiable <interval map> is uniformly expanding when a constant $\lambda>1$ bounds the derivative on every smooth branch:
$$
|T'(x)|\geq\lambda.
$$
With a finite partition into monotone branches, a point's itinerary records which branch contains each iterate. A length-$n$ cylinder consists of points with the same first $n$ branch labels. Wherever this composition is smooth, the <chain rule> gives $|(T^n)'|\geq\lambda^n$.
The <mean value theorem> then bounds the diameter of a cylinder by the length of the original interval times $\lambda^{-n}$. Inverse branches therefore determine starting points with exponentially improving spatial precision as more itinerary symbols are specified.
For the doubling map on $[0,1)$, the two branches have slope two and the level-$n$ cylinders are half-open intervals of length $2^{-n}$. Their labels are the first $n$ digits of the <binary expansion>. Endpoint conventions matter for unique coding, but not for the cylinder-length estimate.
Expansion describes the map's geometry. A <symbolic potential> separately specifies statistical weights, so expansion alone does not make every <Gibbs measure> absolutely continuous. The <geometric potential of an expanding map> uses the inverse derivative to select weights adapted to interval length.
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