Thermodynamic formalism

Thermodynamic formalism studies weighted orbit sums, invariant measures and geometric scaling through analogues of partition functions. A symbolic potential assigns a weight to each orbit segment; its asymptotic logarithmic normalization is topological pressure. The resulting Gibbs measure depends on the potential, so choosing an arbitrary Gibbs law does not automatically select geometric volume.

A Gibbs measure is a probability measure whose finite-block weights or conditional laws are determined by exponential potential sums and their normalization. One must specify whether the convention uses boundary-conditioned finite-volume laws, a normalized one-sided transition kernel, or symbolic Gibbs bounds. For the latter, cylinder probabilities are comparable to $\exp (S_n\varphi -nP(\varphi ))$, where $P$ is topological pressure. These conventions have compatibility requirements; an arbitrary unnormalized one-sided potential need not define consistent finite-horizon conditional probabilities.

For the doubling map, level-$n$ cylinders have Lebesgue measure $2^{-n}$. A Bernoulli symbolic law with probabilities $p,1-p$ is a Gibbs measure for a locally constant potential. On a cylinder with $n$ identical symbols, its density ratio is $(2p{)}^n$. If $p\ne 1/2$, these ratios cannot converge uniformly to a finite positive density. For $p=2/3$, the nonboundary alternating sequence has even-level log ratio $(n/2)\log (8/9)$, which tends to $-\infty$. The geometric Gibbs law is the special case $p=1/2$.

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.

The sequences $a_n+D$ and $a_n-D$ are respectively subadditive and superadditive. Apply Fekete's lemma to the first and its negative to the second; their normalized difference is $2D/n$, so their limits coincide. For full-shift weighted orbit sums, summable variations of the potential bound the energy change at a concatenation boundary, giving this estimate for the logarithm of the partition function. A fixed initial symbol or fixed remote tail changes only a bounded normalization and not the limit.

A symbolic potential assigns a real weight to an infinite symbolic sequence. Its orbit sum gives the energy in a weighted partition function. Agreement on the first $r$ symbols defines the variation ${\mathrm{var}}_{r}U$; summable variations control sensitivity to a remote history. Geometric potentials encode interval contraction, while other potentials can assign different statistical weights to the same symbolic dynamics.