Characterizing symmetric spaces by their Lyapunov spectra

We prove that closed negatively curved locally symmetric spaces are characterized up to isometry among all homotopy equivalent negatively curved manifolds by the Lyapunov spectra of the periodic orbits of their geodesic flows. This is done by constructing a new invariant measure for the geodesic flow that we refer to as the horizontal measure. We show that the Lyapunov spectrum of the horizontal measure alone suffices to locally characterize these locally symmetric spaces up to isometry. We associate to the horizontal measure a new invariant, the horizontal dimension. We tie this invariant to extensions of curvature pinching rigidity theorems for complex hyperbolic manifolds to pinching rigidity theorems for the Lyapunov spectrum. Our methods extend to give a rigidity theorem for smooth Anosov flows $f^{t}$ orbit equivalent to the geodesic flow $g^{t}_{X}$ of a closed negatively curved locally symmetric space $X$: $f^{t}$ is smoothly orbit equivalent to $g^{t}_{X}$ if and only if its Lyapunov spectra on all periodic orbits are a multiple of the corresponding Lyapunov spectra for $g^{t}_{X}$.Comment: 92 pages. Main result has been improved significantl

A quasi-tree expansion of the Krushkal Polynomial

We introduce a generalization of the Krushkal polynomial to nonorientable surfaces, and prove that this polynomial has a natural quasi-tree expansion. This generalized Krushkal polynomial contains the Bollobas-Riordan polynomial of a (possibly nonorientable) ribbon graph as a specialization. The quasi-tree expansion proven here then extends the recent quasi-tree expansions of the Bollobas-Riordan polynomial deduced in the oriented case by A. Champanerkar et al. and in the more general unoriented case by E. Dewey and F. Vignes-Tourneret. The generalized Krushkal polynomial also contains the Las Vergnas polynomial of a cellulation of a surface as a specialization; we use this fact to deduce a quasi-tree expansion for the Las Vergnas polynomial.No embarg