Recovery of vanishing cycles by log geometry

We first construct compatible actions of the product of the unit interval a nd the unit circle as a monoid on a semi-stable degeneration of pairs and o n the associated log topological spaces. Then we show that the log topologi cal family is locally trivial in piecewise smooth category over the base, i .e.. the associated log topological family recovers the vanishing cycles of the original semi-stable degeneration in the most naive sense. Using this result together with the log Riemann-Hilbert correspondence, we introduce t wo types of integral structure of the variation of mixed Hedge structure as sociated to a semi-stable degeneration of pairs.