Constructing metrics on a 2-torus with a partially prescribed stable norm

A result of Bangert states that the stable norm associated to any Riemannian metric on the 2-torus T (2) is strictly convex. We demonstrate that the space of stable norms associated to metrics on T (2) forms a proper dense subset of the space of strictly convex norms on . In particular, given a strictly convex norm || center dot ||(a) on we construct a sequence of stable norms that converge to || center dot ||(a) in the topology of compact convergence and have the property that for each r > 0 there is an such that || center dot || (j) agrees with || center dot ||(a) on for all j a parts per thousand yen N. Using this result, we are able to derive results on multiplicities which arise in the minimum length spectrum of 2-tori and in the simple length spectrum of hyperbolic tori.