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Equivalence of boundary measures on covering trees of finite graphs. (English) Zbl 0821.58008

On the universal cover of a compact negatively curved Riemannian manifold there exist three natural measure classes: harmonic measure, visibility measure and Patterson measure. It has been conjectured that \(M\) is locally symmetric iff any two of these measure classes coincide.
If \(T\) is the universal covering tree of the finite graph \(G\) the boundary \(\partial T\) of \(T\) is the set of all geodesic rays emanating from its root. On \(\partial T\) one can define again three measures: harmonic measure, visibility measure and Hausdorff measure. By analogy with the case of negatively curved covering manifolds, V. A. Kajmanovich [Ann. Inst. Henri Poincaré, Phys. Théor. 53, No. 4, 361- 393 (1990; Zbl 0725.58026)] asked when two of the three natural measure classes on \(\partial T\) coincide.
The paper under review gives an almost complete answer to this question. First, the author develops formulas for the above measures on \(\partial T\) and shows that Hausdorff measure is equal to Patterson measure. Then he answers Kajmanovich’s question proving that if any two of the three measures are equivalent and the graph \(G\) has no vertices of degree two, then \(G\) is regular (i.e. the degree of every vertex is the same) or biregular. Necessary and sufficient conditions that two of three measure classes of \(G\) coincide are given. The only coincidence of Patterson and harmonic measure classes for graphs with some vertices of degree 2 is not resolved satisfactorily.

MSC:

58C35 Integration on manifolds; measures on manifolds
37A99 Ergodic theory

Citations:

Zbl 0725.58026
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References:

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