EDB — 0ZP

[0ZP]Topics:ultrametric.Prerequisites:[0X0]↺↻.

Fix \(𝜆{\gt}0\). We define the ultrametric space of sequences as in Sec. [0MR]↺↻: let \(I\) be a finite set, of cardinality \(p\); let \(X=I^ℕ\) be the space of sequences; define \(c\) as in eqn. [(9.137)]↺↻; define \(d(x,y)=𝜆^{-c(x,y)}\). We know from exercises [0X6]↺↻ and [0X1]↺↻ that \((X,d)\) is compact.

Show that the dimension of \((X,d)\) is \(\log p/\log 𝜆\).