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A Gibbs measure is determined by its potential, rather than by geometric volume alone. The doubling map $T(x)=2x(\mathrm{mod}1)$ gives a concrete counterexample. Away from dyadic rationals, write $x={\sum }_{j\ge 1}{\xi }_j{2}^{-j}$. Give the binary digits independent probabilities $\mathrm{Pr}({\xi }_j=1)=p$ and $\mathrm{Pr}({\xi }_j=0)=1-p$, with $0<p<1$.

A length-$n$ cylinder set with $k$ ones then has probability $p^k(1-p{)}^{n-k}$. This is the Gibbs measure for the one-symbol potential $\varphi (\xi )=\log p$ when ${\xi }_1=1$ and $\log (1-p)$ otherwise. The normalized topological pressure is zero because the length-$n$ weights sum to $(p+1-p{)}^n=1$.

Let $E_p$ be the set of numbers whose binary expansion has limiting frequency $p$ of the digit one. The strong law of large numbers gives ${\mu }_p(E_p)=1$. Under Lebesgue measure, binary digits are independent fair digits: every length-$n$ cylinder has length $2^{-n}$. The same strong law of large numbers therefore gives digit frequency $1/2$ almost surely. For $p\ne 1/2$, it follows that $\mathrm{Leb}(E_p)=0$.

Thus ${\mu }_p$ and Lebesgue measure are mutually singular measures. The ambiguity of dyadic expansions affects only a countable set with zero mass under both measures. The geometric choice is precisely $p=1/2$, when cylinder probabilities equal cylinder lengths.

Comparing ratios ${\mu }_p(C)/\mathrm{Leb}(C)$ on a few cylinders would be a weaker argument. Failure of uniform convergence to a positive bounded density does not exclude every possible integrable density; the full-measure set $E_p$ proves actual singularity.

The Feigenbaum geometric potential is a bounded negative symbolic potential $U$ encoding the lengths of nested intervals of the Feigenbaum attractor. In an appropriate binary coding,and changing only the distant tail of an address changes $U$ by an exponentially small amount. For odd interval labels, a sum along reversed address prefixes satisfieswith a bound independent of the depth and address. Even-label intervals have lengths comparable to neighboring odd-label intervals. These uniform estimates connect geometric lengths with thermodynamic formalism.^[ref][ref]

The bounded logarithmic error is important. For fixed real $t$, exponentiation givesAfter summing and taking $n^{-1}\log$, the error is at most $|t|C/n$, which vanishes. Thus geometric partition sums and symbolic partition sums give the same limiting topological pressure. The almost-additive partition-function limit supplies existence under the bounded-distortion hypotheses; bounded negative potentials give a strictly decreasing pressure.

This potential measures geometric contraction across partition levels. It is not an unbounded derivative potential evaluated at the critical point, where $g^{\prime }(0)=0$, and partition depth is distinct from elapsed orbit time.

A Gibbs measure gives probabilities controlled by exponential weights from a potential. The precise convention must be specified: finite-volume conditional distributions and symbolic orbit-weight bounds are related formulations with different hypotheses.

For a finite-alphabet full shift, a standard symbolic Gibbs condition for a probability measure $\mu$ is that a constant $C\ge 1$ satisfyfor every length-$n$ word $w$ and every $x$ in its cylinder set $[w]$. Here $S_n\varphi (x)={\sum }_{j=0}^{n-1}\varphi ({\sigma }^{j}x)$, and $P(\varphi )$ is topological pressure. The same constant must work for every word and every length. Summable variations make the orbit-weight comparison within one cylinder uniform. In applications to equilibrium states, shift invariance is also required.

For a one-symbol potential $\varphi (x)=\log p_{x_0}$, where every $p_a>0$ and ${\sum }_a{p}_a=1$, the independent-symbol law hasThe bound is exact with $C=1$. Equal symbol probabilities and unequal symbol probabilities are both Gibbs laws, although their geometric images can be very different: an arbitrary Gibbs measure need not be absolutely continuous.

An arbitrary unnormalized list of one-sided weights need not give consistent conditional probabilities for every finite horizon. Normalization and compatibility are part of constructing the measure, rather than consequences of calling the weights Gibbs.

Use the geometric properties of the Feigenbaum geometric potential: for some constants $0<b\le B<\infty$,Here ${\mathrm{var}}_r$ means variation between sequences agreeing in their first $r$ symbols. In particular $D={\sum }_{r\ge 1}{\mathrm{var}}_{r}U<\infty$. These are the standard bounds for the Feigenbaum geometric potential; the symbolic potential is not the singular expression $-\log |g^{\prime }|$ at a quadratic critical point.

First allow all $n$ spins and use a fixed remote tail $\eta$, writing $Z_n(\gamma ;\eta )$ for the weighted partition function. In a concatenated block of lengths $n$ and $m$, replacing the history before the second block by $\eta$ changes its energy by at most $D$: the $j$th summand has the same first $j$ symbols, so its difference is at most ${\mathrm{var}}_{j}U$. ThereforeFor $a_n=\log Z_n$, the almost-additive partition-function limit follows explicitly: $a_n+|\gamma |D$ is subadditive and $a_n-|\gamma |D$ superadditive, so Fekete's lemma gives the same finite limit for both after division by $n$. The finiteness follows from bounded $U$ and the $2^n$ words.

Changing the fixed remote tail also changes each full energy by at most $D$. The question fixes the first spin to one and sums the remaining $n-1$ spins; its partition function is $e^{\gamma U(1,0,0,\dots )}{Z}_{n-1}(\gamma ;(1,0,0,\dots ))$. The tail comparison and bounded first contribution show that its normalized logarithm has the same limit. Thus the requested $f(\gamma )$ exists and equals the topological pressure of $\gamma U$.

For any $\delta >0$, every length-$n$ energy lies between $-Bn$ and $-bn$, henceTaking normalized logarithms and limits givesSo $f$ is Lipschitz continuous and strictly decreasing. For each finite $n$, differentiation of the log-sum gives $f_n^{\prime \prime }(\gamma )=n^{-1}{\mathrm{Var}}_{n,\gamma }(H_n)\ge 0$. Thus $f_n$ is convex, and passing to its finite pointwise limit proves that $f$ is convex too.

At zero weight the prescribed sum has $2^{n-1}$ terms, so $f(0)=\log 2$. For $\gamma >0$ the preceding bounds give $\log 2-B\gamma \le f(\gamma )\le \log 2-b\gamma$. The function therefore becomes negative at sufficiently large positive $\gamma$. Continuity and strict decrease prove

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.

For a finite full shift with alphabet $\mathcal{A}$ and a continuous potential $\varphi$ of summable variations, topological pressure is the limiting logarithmic weight per step:Here $w\eta$ concatenates a finite word and a fixed reference tail, and $\sigma$ is the left shift. Summable variations bound the effect of changing that tail by a constant independent of $n$. The almost-additive partition-function limit consequently proves existence and tail independence.

For a fixed potential $U$, the function $\gamma \mapsto P(\gamma U)$ is convex. At finite $n$, differentiating $n^{-1}\log Z_n(\gamma U)$ twice gives the variance of the orbit sum under its normalized weights, divided by $n$, which is nonnegative. Passing to the limit preserves convexity.

If $-B\le U\le -b<0$, then for $\delta >0$ each orbit weight is multiplied by a factor between $e^{-nB\delta }$ and $e^{-nb\delta }$. HencePressure is therefore strictly decreasing and Lipschitz continuous. If the alphabet has $m\ge 2$ symbols, $P(0)=\log m>0$ and $P(\gamma U)\to -\infty$ as $\gamma \to +\infty$. The intermediate value theorem then gives exactly one positive zero. For a one-symbol alphabet the zero is instead at $\gamma =0$; the alphabet-size qualification is essential.

For example, the constant potential $U=-c$, with $c>0$, has $Z_n(\gamma U)=m^n{e}^{-n\gamma c}$ and $P(\gamma U)=\log m-\gamma c$. Its zero is $(\log m)/c$, making the role of alphabet size explicit.