Spectral degeneracy and escape dynamics for intermittent maps with a hole

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron-Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds, and demonstrate $L^1$ convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.Comment: 34 page

Endothelial cells regulate p53-dependent apoptosis of neural progenitors after irradiation

Endothelial cells represent an important component of the neurogenic niche and may regulate self-renewal and differentiation of neural progenitor cells (NPCs). Whether they have a role in determining the apoptotic fate of NPCs after stress or injury is unclear. NPCs are known to undergo p53-dependent apoptosis after ionizing radiation, whereas endothelial cell apoptosis after irradiation is dependent on membrane acid sphingomyelinase (ASMase) and is abrogated in sphingomyelin phosphodiesterase 1 (smpd1)- (gene that encodes ASMase) deficient mice. Here we found that p53-dependent apoptosis of NPCs in vivo after irradiation was inhibited in smpd1-deficient mice. NPCs cultured from mice, wild type (+/+) or knockout (−/−), of the smpd1 gene, however, demonstrated no difference in apoptosis radiosensitivity. NPCs transplanted into the hippocampus of smpd1−/− mice were protected against apoptosis after irradiation compared with those transplanted into smpd1+/+ mice. Intravenous administration of basic fibroblast growth factor, which does not cross the blood–brain barrier, known to protect endothelial cells against apoptosis after irradiation also attenuated the apoptotic response of NPCs. These findings provide evidence that endothelial cells may regulate p53-dependent apoptosis of NPCs after genotoxic stress and add support to an important role of endothelial cells in regulating apoptosis of NPCs after injury or in disease