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

Abstract

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 $${{\mathbb R}^2}$$ . In particular, given a strictly convex norm || · ||∞ on $${{\mathbb R}^2}$$ we construct a sequence $${\langle {\| \cdot \|}_j \rangle_{j=1}^{\infty}}$$ of stable norms that converge to || · ||∞ in the topology of compact convergence and have the property that for each r > 0 there is an $${N \equiv N(r)}$$ such that || · || j agrees with || · ||∞ on $${{\mathbb Z}^2 \cap \{(a,b) : a^2 + b^2 \leq r \}}$$ for all j ≥ 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.