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

Question

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

Answer 1

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.

As an alternative to constructing a measure by hand like this, your question is related to so-called multi-fractal formalism and rotation theory. You should be able to find general statements which answer your question, but I am not able to give a specific reference.