# A unique equilibrium state which does not have Gibbs property

Let $$T:\Sigma \rightarrow \Sigma$$ be a topologically mixing subshift of finite type and let $$f:\Sigma \rightarrow \mathbb{R}$$ be a continuous functions over $$(T, \Sigma)$$. Assume that there is a unique equilibrium measure $$\mu$$ for $$f$$ because of some reason.

$$\textit{Question}:$$ Does $$\mu$$ necessarily have Gibbs property?

I guess the answer is no, but I can't find a reference.

The measure $$\mu$$ does not necessarily have the Gibbs property. In fact, it has the Gibbs property if and only if $$f$$ has the Bowen property: $$\sup_n \sup \{ |S_n f(x) - S_n f(y)| : x_1 \dots x_n = y_1 \dots y_n \} < \infty$$. Every such $$f$$ has a unique equilibrium measure, but there are some potentials without the Bowen property that still have unique equilibrium measures.

$$\mu$$ Gibbs iff $$f$$ Bowen. The Gibbs property requires that there be $$K>0$$ such that for every $$x\in \Sigma$$ we have $$K^{-1}\leq \frac{\mu[x_1\dots x_n]}{e^{-nP(f) + S_nf(x)}} \leq K.$$ Given $$x,y \in \Sigma$$ with $$x_1\dots x_n = y_1 \dots y_n$$, the only quantity in the corresponding inequalities that can vary is $$S_n f$$, and comparing them gives $$K^{-2} \leq e^{S_n f(x) - S_n f(y)} \leq K^2.$$ Thus $$|S_n f(x) - S_n f(y)| \leq 2\log K$$, which proves the Bowen property. The other direction is classical; see

Bowen, Rufus, Some systems with unique equilibrium states, Math. Syst. Theory 8(1974), 193-202 (1975). ZBL0299.54031.

which gives a more general result (expansive systems with specification, which includes mixing SFTs).

An example of a non-Bowen potential that has a unique equilibrium state.

Hofbauer, Franz, Examples for the nonuniqueness of the equilibrium state, Trans. Am. Math. Soc. 228, 223-241 (1977). ZBL0355.28010.

The example there is the full shift on two symbols 0,1, and the potential is $$f(x) = a_k$$ whenever $$x = 1^k 0\dots$$, where $$a_k$$ is a sequence of real numbers converging to $$0$$. (Also $$f(1^\infty) = 0$$.) Writing $$s_k = a_0 + \cdots + a_k$$, the table on page 239 of that paper is useful. The potential $$f$$ has the Bowen property iff $$\sum a_k$$ converges, but there are examples where $$\sum a_k$$ diverges and $$f$$ still has a unique equilibrium measure.

It is often the case that unique equilibrium measures, including the ones in Hofbauer's paper, satisfy a "non-uniform" Gibbs property: see

Climenhaga, Vaughn; Thompson, Daniel J., Equilibrium states beyond specification and the Bowen property, J. Lond. Math. Soc., II. Ser. 87, No. 2, 401-427 (2013). ZBL1276.37023.