Ergodic dichotomy for subspace flows in higher rank

In rank one, the Hopf-Tsuji-Sullivan dichotomy theorem states the dichotomy for the ergodicity of geodesic flow in terms of the divergence/convergence of the Poincaré series at the critical exponent. Burger-Landesberg-Lee-Oh initiated the study of higher rank version of the Hopf-Tsuji-Sullivan dichotomy for a one-dimensional diagonal flow and the associated Poincaré series. In this talk, we discuss the Hopf-Tsuji-Sullivan dichotomy for higher dimensional flows which we call subspace flows of Weyl chamber flows. Just like the dimension dichotomy for Brownian motions in $\mathbb{R}^n$, the codimension dichotomy of the flow occurs for Anosov homogeneous spaces. This is based on joint work with Hee Oh and Yahui (Amy) Wang.