$\DeclareMathOperator{\top}{\mathrm{top}}$Let $(\Sigma, T)$ be a topologically mixing subshift of finite type and $f:\Sigma \to \mathbb{R}$ be a Hölder continuous map. Let $$K(\alpha)=\Big\{x\in \Sigma, \lim_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1}f(T^{i}(x))=\alpha\Big\}$$ be a level set. One often finds $\alpha \mapsto h_{\top}(K(\alpha))$, where $h_{\top}$ is topological entropy is the sense of Bowen, is the Legendre transform of the pressure function $t\mapsto P(tf)$. It is also easy to show that $\alpha \mapsto h_{\top}(K(\alpha))$ is a concave function.

$\textbf{Problem:}$ I want to understand why the function $h_{\top}(K(\alpha))$ achieves a positive value somewhere and also why the function is nonnegative.

Remark: I probably forget to mention some assumptions ensuring that the above assertion is true; please consider the question under the assumptions that makes sense.