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Resonances for non-compact locally symmetric spaces

Dynamical resonances, also called Pollicott-Ruelle resonances or classical resonances, are a spectral invariant associated with chaotic dynamical systems, formed by a discrete set of complex numbers. For example, if $M$ is a compact Riemannian manifold with negative sectional curvatures, then the geodesic flow $\varphi_t$ on the unit sphere bundle $SM\subset TM$ over $M$ is an example of such a chaotic system. Here chaotic refers to the so-called Anosov property, which says that the subbundle $X^\perp\subset T(SM)$ formed by all vectors that are fiber-wise transverse to the generating vector field $X$ of the geodesic flow splits into a direct sum of two continuous flow-invariant subbundles $E_+$ and $E_-$ on which the derivative $D\varphi_t$ acts exponentially contracting and expanding as $t\to +\infty$, respectively. If $M$ is hyperbolic or, more generally, a Riemannian locally symmetric space of rank $1$, the geodesic flow on the unit sphere bundle over $M$ can be described algebraically in a very convenient way, and the definition and study of dynamical resonances in this setting has been very fruitful in recent years. In particular, the theory has been generalized to convex cocompact hyperbolic manifolds and also to more general non-compact situations. A very recent new direction of research consists in the focus on locally symmetric spaces of higher rank $k>1$, where one studies a multi-temporal flow $\varphi_{t_1,\ldots, t_k}$ with which one would like to associate a multi-dimensional spectrum of dynamical resonances formed by a discrete set in $\mathbb C^k$. So far, this has only been achieved in the cocompact case. A main goal of the project Resonances for non-compact locally symmetric spaces is the generalization to spaces arising from Anosov representations. These spaces lend themselves as promising objects of study due to their good properties regarding compactifications.

Team Members

Dr. Benjamin Delarue