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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.

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    $\begingroup$ The version of $h_\text{top}$ that I am most familiar with is applied to compact sets $K$. In that case in the shift setting, it is just the exponential growth rate of the number of blocks: if $c(n)$ denotes the number of blocks of length $n$ that appear in $K$, $h_\text{top}(K)$ is just the limit of $\frac 1n\log c(n)$. That is, $c(n)\approx e^{h_\text{top}n}$. I believe this is very close to a box dimension. On the other hand, your $K_\alpha$'s are not closed. As you probably know, Hausdorff dimension is bounded above by box dimension because you are have much more freedom in your covers $\endgroup$ Apr 20, 2023 at 12:31
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    $\begingroup$ arxiv.org/abs/1702.04394 I believe this is relevant $\endgroup$
    – Ville Salo
    Apr 20, 2023 at 18:08
  • $\begingroup$ @AnthonyQuas Thank you very much $\endgroup$
    – Adam
    Apr 21, 2023 at 8:43
  • $\begingroup$ @VilleSalo Thank you very much $\endgroup$
    – Adam
    Apr 21, 2023 at 8:43

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The following doesn't necessarily answer your question about how to intuitively interpret entropy and dimension, but it does address the relationship between them, at least in the symbolic setting.

In a one-sided shift space -- whether or not it is an SFT -- topological entropy (in the sense of Bowen's 1973 paper) is identical to Hausdorff dimension, at least up to a scaling constant. The reason for this is that in one-sided shift spaces, metric balls (which are used in the definition of Hausdorff dimension) are identical to dynamical/Bowen balls (which are used in the definition of topological entropy). Indeed, if the metric is defined by $d(x,y) = \theta^n$, where $\theta\in (0,1)$ is a fixed parameter and $n=n(x,y)$ is the smallest index such that $x_n \neq y_n$, then given any $x$ in the shift space and any $n\in \mathbb{N}$, if we write $w = x_1 \cdots x_n$ then the cylinder $[w]$ coincides with both the metric ball $B(x,r)$ for any $r\in (\theta^{n+1},\theta^n]$, and the Bowen ball $B_n(x,\epsilon)$ for any $\epsilon \in (\theta^2,\theta]$. Using this fact you can work directly from the definitions of topological entropy and Hausdorff dimension and quickly show that they agree (up to multiplication by $|\log\theta|$ or some constant like that).

I believe this is all described in the paper by Stephen Simpson that Ville Salo's comment linked to, but it seemed worth spelling out the main parts of the story here.

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    $\begingroup$ Vaughn: can I get you to expand your helpful answer here to address a possible confusion of mine. In my comment on the q, I was suggesting that topological entropy is closer to box dimension than Hausdorff dimension, since with topological entropy you're essentially covering by cylinders all of the same length. Am I missing something? Or is there a reason that for these dynamical sets, Hausdorff dimension and box dimension agree? $\endgroup$ Apr 21, 2023 at 11:58
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    $\begingroup$ @AnthonyQuas , it goes back to Furstenberg, in his seminal Disjoinetness paper, where he showed that for 1-sided subshifts, the Minkowski dimension and Hausdorff dimension coincide. Now the question is about how good of an encoding one gets. $\endgroup$
    – Asaf
    Apr 21, 2023 at 13:26
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    $\begingroup$ Anthony: There's some ambiguous terminology at work here. The most common use of the term "topological entropy" is indeed as you describe: count the number of cylinders/Bowen balls all of the same order, then take the growth rate as that order increases. And in a shift space, that quantity is proportional to the box/Minkowski dimension by exactly the reasoning in my answer. But Bowen's 1973 TAMS paper introduced a different definition of topological entropy, analogous to Hausdorff dimension, which is given not as a growth rate but as a critical parameter... $\endgroup$ Apr 21, 2023 at 14:50
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    $\begingroup$ ... that is, given $h\geq 0$ and a subset $Z$ of the shift space you define $m_h(Z) = \lim_{N\to\infty} \inf \sum_{w\in E} e^{|w|h}$, where the infimum is over all collections $E$ of finite words with the property that $Z \subset \bigcup_{w\in E} [w]$. Then the topological entropy ("in the sense of Bowen" as some people write) is the unique value of $h$ at which $m_h(Z)$ jumps from $\infty$ to $0$. $\endgroup$ Apr 21, 2023 at 14:53
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    $\begingroup$ If $Z$ is compact and shift-invariant then the two definitions of topological entropy agree. But in general they can take different values, as you know well by considering Hausdorff and box dimension. $\endgroup$ Apr 21, 2023 at 14:53

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