I have been reading some results about multifractal formalism. I noticed that some results were proved for the Hausdorff dimension and some results for the topological entropy (in the sense of Bowen). I am familiar with both definitions, which are similar to each other, but I don't have any images of them in mind or a good intuition of them. Could you give me a good intuition or image about them? Thanks in advance.
For instance, we consider the following system: Let $T:\Sigma \to \Sigma$ be a topologically mixing subshift of finite type and $f:\Sigma \to \mathbb{R}$ be a Holder continuous function. Let $K_\alpha$ be the set where $\lim_{n\to \infty}\frac{1}{n}S_{n}f(x)$ exists and is equal to $\alpha$. I want to know how I should imagine $h_{\text{top}}(K_{\alpha})$ and $\dim_{H}(K_{\alpha})$ in my mind.