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Is it true that for a given nonnegative number, there exists a measure-theoretical entropy value (supremum of entropies of all partitions under a measure-preserving transformation) that equals this number?

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Take a shift on $G^{\mathbb Z}$ where $G$ is a finite group. The entropy will be $\log |G|$ (let's say logarithm with base $2$). So if you choose $G$ such that $|G|=2^n$ (for $n$ your favourite positive integer) you have the example you are looking for. –  Simone Virili Nov 12 '12 at 21:14
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In general, using Cantor sets constructions, one can find sub-systems of the shift system of any given entropy between 0 and $\log(n)$. I believe this appears in Furstenberg's disjointness paper for example, although the construction is obviously much older than the paper. This is not a real research question. –  Asaf Nov 12 '12 at 22:11
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Use the intermediate value theorem for Bernoulli measures on $\lbrace 1,\ldots,n\rbrace^{\mathbb Z}$. –  Anthony Quas Nov 12 '12 at 23:58
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2 Answers 2

There is also a paper of Grillenberger, where You can find the construction http://link.springer.com/article/10.1007%2FBF00537161?LI=true

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There are two funny approaches:

  1. Let $S_2$ be a surface of genus 2 with constant curvature $-1$. Then the geodesic flow $\phi_t$ on the unit tangent bundle preserves the Liouville measure $\mu$ and has positive metric entropy: $h_\mu(\phi_t)=|t|\cdot h_\mu(\phi_1)$ with $h_\mu(\phi_1)>0$. So for any given number $r$, just pick $t=r/h_\mu(\phi_1)$. In fact any flow with positive entropy works.

  2. We use the affineness of measure-theoretical entropy $h_{\bullet}(f):\mathcal{M}(f)\to \mathbb{R},\mu\mapsto h_\mu(f)$. Pick a large integer $n\ge e^r$ and consider the full shift $\sigma$ on $\lbrace 1,\cdots,n\rbrace^{\mathbb{Z}}$. Then $\mu=(1/n,\cdots,1/n)^{\oplus\mathbb{Z}}$ has entropy $h_\mu(\sigma)=\log n>r$. Let ${\bf 1}=(1111\cdots)$. Then the enropy of $\mu_p=p\mu+(1-p){\delta_{\mathbf{1}}}$ has entropy $p\log n$. So we set $p=r/\log n$.

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