## View abstract

### Session S33 - Spectral Geometry

Tuesday, July 13, 15:00 ~ 15:20 UTC-3

## Scarring of quasimodes on hyperbolic manifolds

### Lior Silberman

#### The University of British Columbia, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak2911786805ab6ea9b6ca101d042d8ceb').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy2911786805ab6ea9b6ca101d042d8ceb = 'l&#105;&#111;r' + '&#64;'; addy2911786805ab6ea9b6ca101d042d8ceb = addy2911786805ab6ea9b6ca101d042d8ceb + 'm&#97;th' + '&#46;' + '&#117;bc' + '&#46;' + 'c&#97;'; var addy_text2911786805ab6ea9b6ca101d042d8ceb = 'l&#105;&#111;r' + '&#64;' + 'm&#97;th' + '&#46;' + '&#117;bc' + '&#46;' + 'c&#97;';document.getElementById('cloak2911786805ab6ea9b6ca101d042d8ceb').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy2911786805ab6ea9b6ca101d042d8ceb + '\'>'+addy_text2911786805ab6ea9b6ca101d042d8ceb+'<\/a>';

Let $M$ be a compact hyperbolic manifold. The entropy bounds of Anantharaman et al. restrict the possible invariant measures on $T^1 M$ that can be quantum limits of sequences of eigenfunctions. Weaker versions of the entropy bounds also apply to approximate eigenfuctions ("log-scale quasimodes"), so it is interesting to construct such approximate eigenfunctions which converges to singular measures.

Generalizing work of Brooks (hyperbolic surfaces) and Eswarathasan--Nonnenmacher (hyperbolic geodesics on Riemannian surfaces) we construct sequences of quasimodes on $M$ converging to totally geodesic submanifolds. A diagonal argument then realizes every invariant measure are a limit of quasimodes of fixed logarithmic width.

Joint work with Suresh Eswarathasan (Dalhousie University, Canada).