= Symbolic potential
{title2=$U:K^{\mathbb N}\to\mathbb R$}
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 sum
$$
S_nU(x)=\sum_{j=0}^{n-1}U(\sigma^jx)
$$
weights 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 $r$ is
$$
\operatorname{var}_rU=\sup\{|U(x)-U(y)|:x_0=y_0,\ldots,x_{r-1}=y_{r-1}\}.
$$
Summable variations mean $\sum_{r\geq1}\operatorname{var}_rU<\infty$. If $x$ and $y$ agree on their first $n$ symbols, their shifted sequences agree for $n-j$ symbols at step $j$. Therefore
$$
|S_nU(x)-S_nU(y)|\leq\sum_{r=1}^n\operatorname{var}_rU.
$$
The 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 $r\geq1$ 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.
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