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I have a research project involving ultrametric spaces, and there are some facts that I use but have a hard time finding explicitely in the literature, although I know that some of them are folklore (for example, an ultrametric space can be described as the set of leaves of a tree, endowed with the induced metric).

I would like to know whether there is a book or comprehensive survey paper on the geometry and structure of ultrametric spaces.

An important point: I am interested in purely metric spaces, without algebraic structure (I did find books on analysis in non-Archimedean fields, which are too focused on this case). I can restrict to compact spaces, but not to finite ones.

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    $\begingroup$ rmp.aps.org/abstract/RMP/v58/i3/p765_1 $\endgroup$ Apr 8, 2013 at 16:00
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    $\begingroup$ Automata, dynamical systems and infinite groups, with V.V.Nekrashevich, V.I.Sushchanskii, Proc. Steklov Inst. Math. v.231 (2000), 134-214 gives the description of ultrametric spaces in terms of trees. $\endgroup$ Apr 8, 2013 at 16:43
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    $\begingroup$ You might also be interested in the use of ultrametric spaces in the functorial approach to hierarchical clustering algorithms: see, e.g., Carlsson and Memoli's papers. $\endgroup$ Apr 8, 2013 at 19:54
  • $\begingroup$ You say you do not want any algebraic structure, so may be infinite-dimensional vector spaces are not ok for you. But otherwise, there is a beautiful small book by Peter Schneider called "Non-archimedean functional analysis" (a Springer Monograph in Math.) where he gives a lot of results. In contrast to the books you mention, he does not focus on fields (1-dim'l case, if you want), neither on complete spaces. But all his spaces are vector spaces... $\endgroup$ Apr 9, 2013 at 1:47
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    $\begingroup$ Lionel Nguyen Van Thé's thesis was all about structural properties of ultrametric spaces. I bet some of his papers and/or the references in there will prove helpful for you - latp.univ-mrs.fr/~lionel/englishindex.html $\endgroup$ Apr 22, 2013 at 19:37

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Maybe the following paper and references in there:

  • MR2026825 (2005j:54018) Lemin, Alex J.(RS-MSUCE) The category of ultrametric spaces is isomorphic to the category of complete, atomic, tree-like, and real graduated lattices LAT∗. (English summary) Algebra Universalis 50 (2003), no. 1, 35–49.
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What about

MR2093482 (2005m:57001) Hughes, Bruce. Trees and ultrametric spaces: a categorical equivalence. (English summary). Adv. Math. 189 (2004), no. 1, 148–191.

It's not a survey, but it deals with the correspondence between ultrametric spaces and the ends of a tree.

From the review on mathscinet:

`The author studies the correspondence between infinite trees and ultrametric spaces arising from the end space of the tree. The main theorem is the existence of an equivalence defined from the category of geodesically complete, rooted $\mathbb{R}$-trees and equivalence classes of isometries at infinity, to the category of complete ultrametric spaces of finite diameter and local similarity equivalences.'

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  • $\begingroup$ Thanks for the suggestion, but as I did my homework, I knew the existence of this paper; the point is that the correspondence itself is said to be "well-known" in the abstract, so this is not the primary reference for this particular point. Besides, its purpose is to describe categorically this correspondence, so there seems to be quite little about other structural properties of ultrametric spaces. This comment applies equally well to Lemin's paper cited by Peter Michor. $\endgroup$ Apr 23, 2013 at 11:02

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