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A piecewise differentiable interval map is uniformly expanding when a constant $\lambda >1$ bounds the derivative on every smooth branch: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 $|(T^n{)}^{\prime }|\ge {\lambda }^n$.

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.

For a differentiable uniformly expanding interval map, the geometric potential of an expanding map isThe exponential of its length-$n$ orbit sum is the reciprocal derivative magnitude $1/|(T^n{)}^{\prime }(x)|$. 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:For a finite smooth full-branch expanding map, standard bounded-distortion hypotheses give a positive regular fixed density $h$, normalized by $\int hdx=1$. The identity $\mathcal{L}h=h$ 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 andThe 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.

A qualification is necessary: an arbitrary symbolic Gibbs measure does not satisfy the claim. Consider the doubling map with its two binary branches and a Bernoulli symbolic Gibbs measure with probabilities $2/3$ and $1/3$. This is the Gibbs law for a locally constant symbolic potential, while every level-$n$ interval has length $2^{-n}$. Along the nonboundary alternating itinerary, for even $n$,No finite $g$ can approximate these $g_n$ with a decaying uniform error. This proves the reusable fact that an arbitrary Gibbs measure need not be absolutely continuous.

For the intended geometric version, take the Gibbs measure for the geometric potential of an expanding map $-\log |T^{\prime }|$, with the standard sufficiently smooth finite full-branch hypotheses. The allowed transfer-operator result supplies a strictly positive Lipschitz continuous invariant probability density function $h$, normalized by $\int h=1$, and its symbolic weights of cylinder sets are $\nu (C_w)={\int }_{{\Delta }_w}h(x)dx$. State the needed regularity explicitly: $h\ge h_{\ast }>0$ and $|h(x)-h(y)|\le L_h|x-y|$. Expansion gives $\mathrm{diam}{\Delta }_w\le {\lambda }^{-n}$.

Let $\pi (ϵ)$ denote the point coded by the infinite sequence, and define $g(ϵ)=\log h(\pi (ϵ))$. The expression defining $g_n$ is the logarithm of the average of $h$ over the cylinder. For $x=\pi (ϵ)\in {\Delta }_w$,Since the logarithm is Lipschitz continuous with constant $1/h_{\ast }$ on this positive range, the cylinder averages of an invariant density giveThis proves the intended estimate. Expansion alone and an unspecified Gibbs symbolic potential are insufficient; the geometric choice and the regularity producing a positive Lipschitz continuous probability density function are substantive hypotheses.

A symbolic potential is a real-valued function on a symbolic sequence space, for example $U:{\mathcal{A}}^{\mathbb{N}}\to \mathbb{R}$ on a finite full shift. Its orbit sumweights a length-$n$ orbit segment in a partition function. Here $\sigma$ is the left shift. The potential can encode geometric contraction or a chosen statistical preference; it is not determined by the alphabet alone.

Its variation at depth $r$ isSummable variations mean ${\sum }_{r\ge 1}{\mathrm{var}}_{r}U<\infty$. If $x$ and $y$ agree on their first $n$ symbols, their shifted sequences agree for $n-j$ symbols at step $j$. ThereforeThe right side stays bounded independently of $n$. This uniform orbit-weight distortion is what allows a reference tail to be changed without changing topological pressure.

For a potential depending only on $x_0$, every variation with $r\ge 1$ is zero. The choice $U(x)=\log p_{x_0}$ generates independent-symbol weights, while a geometric potential of an expanding map instead describes inverse-branch contraction.

Thermodynamic formalism studies weighted trajectories through partition functions, topological pressure and invariant measures. A symbolic potential $\varphi$ assigns an orbit segment the weight $e^{S_n\varphi }$, where $S_n\varphi$ is its accumulated potential. Summing these weights gives $Z_n(\varphi )$; the limiting logarithmic normalization per step is the pressure.

A one-symbol example shows the mechanism. On a two-symbol full shift, let $\varphi (0)=a$ and $\varphi (1)=b$. ThenNormalizing gives independent symbol probabilities $p_0=e^a/(e^a+e^b)$ and $p_1=e^b/(e^a+e^b)$. The associated Gibbs measure is therefore an orbit-weight law, not a uniform choice unless $a=b$.

The same construction can encode geometry. A geometric potential of an expanding map records inverse-branch contraction, while the Feigenbaum geometric potential encodes lengths in a nested partition. Their pressure functions organize spatial scaling and dimension calculations. In that second example, the partition depth is a geometric scale index and should not automatically be identified with elapsed orbit time.

This explains both the usefulness and the limitation of the statistical-mechanics analogy: changing the potential changes which trajectories or geometric pieces receive weight. An arbitrary Gibbs measure need not be absolutely continuous, even when its underlying interval map is expanding.