Ultrametric space

In mathematics, an ultrametric space is a special kind of metric space in which the triangle inequality is replaced with . Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems for ultrametric spaces may seem strange at a first glance, they appear naturally in many applications.

Formal definition

Formally, an ultrametric space is a set of points with an associated distance function (also called a metric)

(where is the set of real numbers), such that for all , one has:

  1. iff
  2. (symmetry)
  3. (strong triangle or ultrametric inequality).

In the case when is a group and is generated by a length function (so that ), the last property can be made stronger using the Krull sharpening[1] to:

with equality if .

We want to prove that if , then the equality occurs if . Without loss of generality, let us assume that . This implies that . But we can also compute . Now, the value of cannot be , for if that is the case, we have contrary to the initial assumption. Thus, , and . Using the initial inequality, we have and therefore .


Even some isosceles triangles cannot exist in an ultrametric space

From the above definition, one can conclude several typical properties of ultrametrics. For example, in an ultrametric space, for all and :

In the following, the concept and notation of an (open) ball is the same as in the article about metric spaces, i.e.


Proving these statements is an instructive exercise.[2] All directly derive from the ultrametric triangle inequality. Note that, by the second statement, a ball may have several center points that have non-zero distance. The intuition behind such seemingly strange effects is that, due to the strong triangle inequality, distances in ultrametrics do not add up.




  1. Planet Math: Ultrametric Triangle Inequality
  2. Stack Exchange: Ultrametric Triangle Inequality
  3. Osipov, Gutkin (2013), "Clustering of periodic orbits in chaotic systems", Nonlinearity (26): 177–200, doi:10.1088/0951-7715/26/1/177.
  4. Leclerc, Bruno (1981), "Description combinatoire des ultramétriques", Centre de Mathématique Sociale. École Pratique des Hautes Études. Mathématiques et Sciences Humaines (in French) (73): 5–37, 127, MR 623034.
  5. Mezard, M; Parisi, G; and Virasoro, M: SPIN GLASS THEORY AND BEYOND, World Scientific, 1986. ISBN 978-9971-5-0116-7
  6. Rammal, R.; Toulouse, G.; Virasoro, M. (1986). "Ultrametricity for physicists". Reviews of Modern Physics. 58 (3): 765–788. doi:10.1103/RevModPhys.58.765. Retrieved 20 June 2011.
  7. Legendre, P. and Legendre, L. 1998. Numerical Ecology. Second English Edition. Developments in Environmental Modelling 20. Elsevier, Amsterdam.

Further reading

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