= Geometric potential of an expanding map
{title2=$\phi=-\log|T'|$}
For a differentiable uniformly <expanding interval map>, the <geometric potential of an expanding map> is
$$
\phi(x)=-\log|T'(x)|.
$$
The exponential of its length-$n$ orbit sum is the reciprocal derivative magnitude $1/|(T^n)'(x)|$. This weight is geometric: an inverse branch contracts interval lengths by that factor.
The corresponding <Perron-Frobenius operator> transports a density by summing contributions from inverse branches:
$$
\mathcal Lh(x)=\sum_{Ty=x}\frac{h(y)}{|T'(y)|}.
$$
For a finite smooth full-branch expanding map, standard bounded-distortion hypotheses give a positive regular fixed density $h$, normalized by $\int h\,dx=1$. The identity $\mathcal Lh=h$ means that $h(x)\,dx$ is an <invariant measure>. The <cylinder averages of an invariant density> explain how the geometric cylinder weights recover that density.
For the doubling map, there are two inverse branches and
$$
\mathcal Lh(x)=\frac12h(x/2)+\frac12h((x+1)/2).
$$
The constant density $h=1$ is fixed, giving <Lebesgue measure>. Choosing an unequal-symbol potential instead produces a different <Gibbs measure>; <an arbitrary Gibbs measure need not be absolutely continuous>. Thus the potential, rather than the symbolic alphabet alone, determines whether the measure agrees with geometric length.
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