Rigidity on integrability of invariant bundles of Anosov maps

In this short survey, we summarize the authors' works about rigidity on integrability of invariant subbundles for Anosov maps and the maps in their homotopic classes. For Anosov diffeomorphisms, we give the equivalent relationship between the joint integrability of strong bundles and the spectral rigidity of center bundles. Based on the previous works, we also prove in the present paper that for a partially hyperbolic diffeomorphism with the one-dimensional center bundle which is homotopic to an Anosov diffeomorphism by a path in the set of partially hyperbolic diffeomorphisms, its stable bundle and unstable bundle are jointly integrable if and only if the center Lyapunov exponents of its periodic points are equal to one of its linearizations. For non-invertible Anosov maps, we build a connection among the integrability of unstable bundles, the rigidity of Lyapunov exponents on the stable bundles, and the existence of topological conjugacy. We also completely classify the conjugacy classes of non-invertible Anosov maps on the torus. Some rigidity issues on the existence of semi-conjugacy are also considered.