A note on a critical problem with natural growth in the gradient

In this paper, we prove the existence of nested sequences of index filtrations for convergent sequences of (admissible) semiflows on a metric space. This result is new even in the context of flows on a locally compact space. The nested index filtration theorem implies the continuation of homology index braids which, in turn, implies the continuation of connection matrices in the infinite-dimensional Conley index theory.