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For the doubling map, level-$n$ cylinders have Lebesgue measure $2^{-n}$. A Bernoulli symbolic law with probabilities $p,1-p$ is a Gibbs measure for a locally constant potential. On a cylinder with $n$ identical symbols, its density ratio is $(2p{)}^n$. If $p\ne 1/2$, these ratios cannot converge uniformly to a finite positive density. For $p=2/3$, the nonboundary alternating sequence has even-level log ratio $(n/2)\log (8/9)$, which tends to $-\infty$. The geometric Gibbs law is the special case $p=1/2$.

A piecewise smooth interval map is uniformly expanding when every smooth branch has derivative magnitude at least $\lambda >1$. A finite branch partition gives a symbolic itinerary and cylinders with diameters at most a constant times ${\lambda }^{-n}$. Expansion alone does not make every symbolic Gibbs measure absolutely continuous; the geometric potential and suitable branch regularity are essential.

For a sufficiently regular uniformly expanding full-branch interval map, the geometric potential assigns the inverse Jacobian determinant weight. Its transfer operator is $\mathcal{L}h(x)={\sum }_{Ty=x}h(y)/|T^{\prime }(y)|$. Under the standard smooth full-branch hypotheses, a positive Lipschitz fixed density $h$ generates the corresponding invariant symbolic Gibbs measure. A different potential generally produces another measure, possibly singular with respect to Lebesgue measure.

We prove uniqueness in the sense of at most one Gibbs measure satisfying the printed conditional distributions. No shift-invariance assumption is needed in advance. Write a history as $\eta =({ϵ}_{s-1},{ϵ}_{s-2},\dots )$. The variation hypothesis makes $U$ continuous and bounded on the finite-alphabet sequence space; let $u_{-}\le U\le u_{+}$ and $V=u_{+}-u_{-}$. The conditional distribution of the first spin, obtained by marginalizing the prescribed two-spin law, must beIn particular $q(a\mid \eta )\ge \delta :=e^{-2V}/k>0$ for every symbol and history. Any two candidate measures have this same time-homogeneous one-step kernel, by conditional marginalization and the tower property.

If $\eta ,{\eta }^{\prime }$ agree in their most recent $r$ symbols, each two-spin energy in this formula changes by at mostBoth numerator and denominator therefore change by factors between $e^{-D{\lambda }^{-r}}$ and $e^{D{\lambda }^{-r}}$. Consequently the total variation distance of the two kernels is at most $e^{2D{\lambda }^{-r}}-1\le B{\lambda }^{-r}$, where $B=e^{2D}-1$. Uniform positivity also bounds it by $1-k\delta$. PutThese bounds are decreasing, strictly below one, and summable. We now give the summable-memory coupling argument rather than assuming a small one-step contraction coefficient.

Starting from any two histories at a remote time, use a maximal coupling of the next-spin laws. If the actual histories have agreed for $r$ consecutive symbols, their probability of disagreement is at most $b_r$. Use shared independent uniform random numbers to construct a dominating match-length process $R$: it increases from $r$ to $r+1$ with probability $1-b_r$ and resets to zero with probability $b_r$. Since $b_r$ decreases, the actual consecutive-match length can be kept at least $R$ at every step. When the dominating process does not reset, the actual spins can be coupled to agree.

A new run from zero has probabilityof never resetting. The product is positive because $b_r<1$ and $\sum b_r<\infty$: its tail logarithm converges absolutely, and its finite initial factors are positive. After each reset a fresh run has the same success probability, so the probability of $m$ failed runs is $(1-p_{\ast }{)}^m$. There are almost surely only finitely many resets. Every failed run has finite duration, and thus the time of the final reset is finite almost surely. Its distribution depends only on $(b_r)$, not on the two starting histories. It follows that the probability of any disagreement in a fixed future block tends uniformly to zero as the block's distance from the initial time tends to infinity.

Now take two candidate Gibbs measures, sample their respective histories before time $-N$, and couple their forward kernels as above. Their finite-block laws really are obtained by these kernels: successive conditional expectations use the common conditional distribution $q$. For any fixed coordinate block $[s,t]$, the coupling inequality for total variation bounds the distance between the two candidate block laws by the dominating probability of a reset at or after step $N+s$. That bound tends to zero as $N\to \infty$. Hence all probabilities of finite cylinder sets agree; cylinder sets generate the product Borel sigma-algebra, so the measures agree.

Thus there is at most one measure satisfying these conditional specifications. There is a separate compatibility issue: for an arbitrary unnormalized symbolic potential the finite-horizon conditional distributions printed in the question need not be consistent as the right endpoint changes. The argument proves the requested uniqueness whenever such a law exists; it does not silently infer existence from the variation estimate. A normalized one-sided kernel or the usual compatible Gibbs specification supplies the standard existence formulation. For a concrete incompatibility, take two symbols and $U(a,\eta )=\log W_{{\eta }_1,a}$ with $W=\left(\begin{array}{cc}1& 1\\ 1& 2\end{array}\right)$. This has zero variation once two symbols agree. From a history ending in symbol one, the first-symbol probability obtained from a two-spin block is $2/5$, but from a three-spin block it is $5/13$. Thus the printed finite-horizon specification can indeed have no measure, even though its uniqueness implication remains valid.

Assume the uniform estimate obtained in the geometric setting of part (i). The limit $g$ is bounded and continuous on the finite-alphabet symbolic space, so $p(x)=e^{g(E(x))}$ is measurable, positive, and bounded. Possible duplicate itineraries at partition endpoints do not affect its Lebesgue measure density.

For a level-$n$ interval ${\Delta }_w$, the estimate givesFix a level-$m$ cylinder $C_v$ and subdivide its interval into level-$n$ intervals, with $n\ge m$. Integrating these inequalities and summing the subcylinder masses yieldsLetting $n\to \infty$ proves the exact equality ${\int }_{{\Delta }_v}pdx=\nu (C_v)$. The same argument on the entire interval gives $\int p=1$. Shrinking cylinders generate the interval Borel sigma-algebra, so the Monotone class theorem identifies $p(x)dx$ with the pushforward ${\pi }_{\ast }\nu$.

The symbolic Gibbs measure in this setting is shift invariant, and coding satisfies $T\circ \pi =\pi \circ \sigma$ away from the endpoint exceptions. These exceptions form a null set: bounded $p$ and shrinking cylinders give zero mass to individual endpoints. Therefore, for every Borel set $B$,Consequentlyis an absolutely continuous invariant measure. This implication uses the uniform cylinder estimate directly and does not rely on naming an invariance theorem in place of the requested argument.

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.

Thermodynamic formalism studies weighted orbit sums, invariant measures and geometric scaling through analogues of partition functions. A symbolic potential assigns a weight to each orbit segment; its asymptotic logarithmic normalization is topological pressure. The resulting Gibbs measure depends on the potential, so choosing an arbitrary Gibbs law does not automatically select geometric volume.