# Lyapunov spectrum($h_{\mathrm{top}}(K(\alpha))$) achieves a positive value somewhere

$$\DeclareMathOperator{\top}{\mathrm{top}}$$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.

What you have here is not a very good definition of topological entropy, I think. I guess your set $$K(\alpha)$$ is not a closed set either, which is not the context where Bowen defined topological entropy. One likely way to define your function is $$\sup\{h(\mu):\int f\,d\mu=\alpha\}$$, where the sup is taken over invariant measures. By compactness, it’s actually a max. With this definition your function is evidently non-negative (since measure-theoretic entropy is non-negative).
Now to show positivity somewhere, let me make the assumption that $$f$$ is Holder continuous. It’s probably not a critical assumption, but it will make life easier. Let $$a=\min\{\int f\,d\mu\}$$ (again taken over invariant measures) and $$A$$, the corresponding maximum. If $$a=A$$, just choose any measure of positive entropy. If $$\alpha\in(a,A)$$, you can build a suitable measure by randomly alternating pieces where the integral is $$a$$ and pieces where the integral is $$A$$ (this guarantees positive entropy). Actually ensuring the integral is precisely $$\alpha$$ is slightly fiddly, but can be done by an intermediate value argument. See for example the lecture notes “Coupling and Splicing” on my web page.