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We define an integer-valued non-degenerate bi-invariant metric (thediscriminant metric) on the universal cover of the identity component of thecontactomorphism group of any contact manifold. This metric has a very simplegeometric definition, based on the notion of discriminant points ofcontactomorphisms. Using generating functions we prove that the discriminantmetric is unbounded for the standard contact structures on R^{2n} x S^1 andRP^{2n+1}. On the other hand we also show by elementary arguments that thediscriminant metric is bounded for the standard contact structures on R^{2n+1}and S^{2n+1}. As an application of these results we get that the contactfragmentation norm is unbounded for R^{2n} x S^1 and RP^{2n+1}. By elaboratingon the construction of the discriminant metric we then define a secondinteger-valued bi-invariant metric, that we call the discriminant oscillationmetric. This second metric is non-degenerate if and only if the contactmanifold is orderable in the sense of Eliashberg and Polterovich and, in thiscase, it is compatible with the partial order. Finally we define thediscriminant and oscillation lengths of a Legendrian isotopy, and prove thatthey are unbounded for T^{\ast}B x S^1 for any closed manifold B, for RP^{2n+1}and for some 3-dimensional circle bundles.

The discriminant and oscillation lengths for contact and Legendrian isotopies