Constructing metrics on a 2-torus with a partially prescribed stable norm |
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Authors: | Eran Makover Hugo Parlier Craig J Sutton |
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Institution: | 1. Department of Mathematics, Central Connecticut State University, New Britain, CT, USA 2. Department of Mathematics, University of Fribourg, Fribourg, Switzerland 3. Department of Mathematics, Dartmouth College, Hanover, NH, 03755, USA
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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. |
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