Polyhedral Billiards, Eigenfunction Concentration and Almost Periodic Control
We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called “pockets”. We prove there are only finitely many immersed periodic tubes missing the pockets and moreover establish a new quantitative estimate for the leng...
Saved in:
Published in | Communications in mathematical physics Vol. 377; no. 3; pp. 2451 - 2487 |
---|---|
Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
2020
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
Cover
Loading…
Summary: | We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called “pockets”. We prove there are only finitely many immersed
periodic tubes
missing the pockets and moreover establish a new quantitative estimate for the lengths of such tubes. This extends well-known results in dimension 2. We then apply these dynamical results to prove a quantitative Laplace eigenfunction mass concentration near the pockets of convex polyhedral billiards. As a technical tool for proving our concentration results on irrational polyhedra, we establish a control-theoretic estimate on a product space with an almost-periodic boundary condition. This extends previously known control estimates for periodic boundary conditions, and seems to be of independent interest. |
---|---|
Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 Communicated by J. Marklof |
ISSN: | 0010-3616 1432-0916 |
DOI: | 10.1007/s00220-020-03741-0 |