= Topological pressure
{title2=$P(\phi)=\lim_n n^{-1}\log\sum_{|w|=n}e^{S_n\phi(w\eta)}$}
For a finite <full shift> with alphabet $\mathcal A$ and a continuous potential $\phi$ of summable variations, <topological pressure> is the limiting logarithmic weight per step:
$$
P(\phi)=\lim_{n\to\infty}\frac1n\log Z_n(\phi),\qquad
Z_n(\phi)=\sum_{w\in\mathcal A^n}\exp\!\left(\sum_{j=0}^{n-1}\phi(\sigma^j(w\eta))\right).
$$
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 $-B\leq U\leq-b<0$, then for $\delta>0$ each orbit weight is multiplied by a factor between $e^{-nB\delta}$ and $e^{-nb\delta}$. Hence
$$
-B\delta\leq P((\gamma+\delta)U)-P(\gamma U)\leq-b\delta.
$$
Pressure is therefore strictly decreasing and <Lipschitz continuous>. If the alphabet has $m\geq2$ 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 $Z_n(\gamma U)=m^ne^{-n\gamma c}$ and $P(\gamma U)=\log m-\gamma c$. Its zero is $(\log m)/c$, making the role of alphabet size explicit.
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