Let $(\Sigma, T)$ be a topologically mixing subshifts of finite type and $f:\Sigma \to \mathbb{R}$ be a Holder continuous.

Let $K(\alpha)=\{x\in \Sigma, \lim_{n\rightarrow \infty}\frac{1}{n}f(T^{i}(x))=\alpha\}$ 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.

$\textbf{Question:}$ 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 such that the above question is true, please conisder the question under the assumptions that makes sence.