Purity for families of Galois representations

We formulate a notion of purity for p-adic big Galois representations and pseudorepresentations of Weil groups of e-adic number fields for e = ≠pp. This is obtained by showing that all powers of the monodromy of any big Galois representation stay "as large as possible" under pure specializations. Using purity for families, we improve a part of the local Langlands correspondence for GL_nin families formulated by Emerton and Helm. The role of purity for families in the study of variation of local Euler factors, local automorphic types along irreducible components, intersection points of irreducible components of p-adic families of automorphic Galois representations is illustrated using the examples of Hida families and eigenvarieties.