 Topological pressure

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.