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Minimal volume entropy for graphs. (English) Zbl 1155.37014

Let \(X\) be a finite connected graph such that the valency denoted by \(k_x+1\), at each vertex \(x\), is at least 3. The author proves that there exists a unique normalized length distance \(d\) minimizing the volume entropy \(h_{vol}(d)\). Moreover, the minimal volume entropy is \[ h_{min}= \frac{1}{2}\sum_{x\in VX} (k_x+1) \log k_x \] and the entropy minimizing length distance \(d=d_{\ell}\) is given by \[ \forall e\in EX, \quad \ell(e)= \frac{\log(k_{i(e)}k_{t(e)})}{\sum_{x\in VX} (k_x+1) \log k_x} \] where \(VX\) (resp. \(EX\)) is the set of verties (resp. oriented edges) of \(X\) and \(i(e)\) (resp. \(t(e)\)) is the initial (resp. terminal) vertex of the edge \(e\).
The author gives also many consequences of this result. In particular, he proves an analogous result for graphs of groups.

MSC:

37B40 Topological entropy
20E08 Groups acting on trees
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References:

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