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The geometric symbolic potential for the Feigenbaum period-doubling map has uniformly negative bounded values and exponentially decreasing variations. Potential sums describe logarithmic lengths of odd-address period-doubling intervals up to a uniform additive error. This is a geometric scaling potential, distinct from an unbounded derivative potential at a critical point. Its pressure is continuous, strictly decreasing and convex by the almost-additive partition-function limit and finite-volume inequalities.

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

For a finite full shift and a potential with summable variations, the displayed symbolic pressure exists and does not depend on the reference tail $\eta$. The almost-additive partition-function limit proves existence. The function $\gamma \mapsto P(\gamma U)$ is convex; finite-volume second derivatives are energy variances divided by block length. If $-B\le U\le -b<0$, its increments for $\delta >0$ lie between $-B\delta$ and $-b\delta$, so it is Lipschitz continuous, strictly decreasing, and has one positive zero.