Semisimplicity, Rigidity, and Unique Ergodicity in Products and Branched Covers of Translation Surfaces

This dissertation studies two aspects of translation-surface dynamics: the structure of natural cocycles over products of strata, and the dynamics of explicit branched-cover constructions. In the first part, I study the Kontsevich--Zorich cocycle and related tensor-power cocycles over products of strata of Abelian differentials, equipped with ergodic actions of subgroups of $(\mathrm{SL}_2\mathbb{R})^n$. I prove Deligne-type semisimplicity results in this setting: the relevant Hodge bundles admit decompositions into $G$-invariant pieces of the form $V_i \otimes_{A_i} W_i$, where the $V_i$ carry compatible variations of Hodge structure, and every $G$-invariant subbundle is obtained from this decomposition by choosing appropriate $A_i$-submodules. I also prove a rigidity result for measurable invariant subbundles on affine invariant manifolds, showing that in affine coordinates the corresponding projection operators are jointly polynomial of homogeneous degree $0$ in the coordinates and the reciprocal area function. The second part studies slit-induced branched double covers of translation surfaces with uniquely ergodic vertical flow. Together with Elizaveta Shuvaeva, we prove a geometric criterion for unique ergodicity based on the existence of embedded neighborhoods of a branch point along a sequence in the Teichmüller orbit. We then show that such neighborhoods occur for almost every slit endpoint under a subsequential embedded-radius hypothesis, and we develop geometric tools based on Delaunay triangulations and pipe cylinders to verify this condition. This yields the following: for almost every choice of a slit, the resulting branched double cover is uniquely ergodic.