Spectral asymptotics of foliated manifolds (Q1331505)

We show that the dilational equivalence classes near zero of the spectral distribution functions for the analytic and combinatorial leafwise Laplacians on a foliated manifold admitting a transverse invariant measure are the same. We show that this dilational equivalence class is invariant under a measure preserving leafwise homotopy equivalence. This is equivalent to the invariance of the dilational equivalence class near infinity of the trace of the leafwise heat kernels.