Dave Constantine
Wesleyan University
Title
Sub-actions and marked length spectrum rigidity for some locally CAT(-1) spaces
Abstract
Let $\phi_t$ be a flow on $X$ and $A:X\to\mathbb{R}$ a Holder function. If $\int_0^T A(\phi_t x)dt \geq V(\phi_T x)-V(x)$ for all $x$ and $T$, $V$ is a sub-action for $A$. A clear necessary condition for the existence of a sub-action is that the integral of $A$ around any closed orbit is nonnegative. Lopes and Thieullen prove that this condition is sufficient for Anosov flows, providing what can be thought of as a positive Livsic theorem. In this talk I'll explain how we generalize their work to geodesic flow on locally CAT(-1) spaces. We then apply this result to certain locally CAT(-1) spaces where marked length spectrum rigidity is known (surface amalgams and Fuchsian building quotients provide some examples). We prove that if there is a marked length spectrum inequality between two spaces, yet their volume is the same, they are in fact isometric, using a proof strategy due to Croke and Dairbekov. This is joint work with Elvin Shrestha and Yandi Wu.
