Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider Bernoulli measures on $X$ only.

Let $f:X\to \mathbb{R}$ be Holder continuous. The measure $\mu$ is a **Gibbs measure** with potential $f$ if there are $C>0$ and $P\in\mathbb{R}$, such that for every infinite sequence $i_1 i_2\ldots$ and all natural $n$,
$$
C^{-1} \le \frac{\mu[i_1\ldots i_n]}{\exp(-nP+f(i)+f(\sigma i)+\cdots+f(\sigma^{n-1}i))} \le C,
$$
where $\sigma$ is the one side shift.

we write $h(\mu)$ for the Kolmogorov-Sinai (metric) entropy of $\mu$.

$$Question$$ Consider a topology on the Gibbs measures via the Holder topology on the potentials.Why is $h(\mu)$ continuous with respect to this topology?

**comment** we know that entropy is upper semi continuous,thus we have to show that why $h(\mu)$ is lower semi continuous.