Back to Search
Start Over
Measure theoretic rigidity for Mumford curves
- Source :
- Ergodic Theory and Dynamical Systems, 33(3), 851. Cambridge University Press, Ergodic Theory and Dynamical Systems. Cambridge University Press
- Publication Year :
- 2013
-
Abstract
- One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincar\'e disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic. In this paper, we find the corresponding statement for Mumford curves, a nonarchimedean analog of Riemann surfaces. In this case, the mere absolute continuity of the boundary map (for Schottky uniformization and the corresponding Patterson-Sullivan measure) only implies isomorphism of the special fibers of the Mumford curves, and the absolute continuity needs to be enhanced by a finite list of conditions on the harmonic measures on the boundary (certain nonarchimedean distributions constructed by Schneider and Teitelbaum) to guarantee an isomorphism of the Mumford curves. The proof combines a generalization of a rigidity theorem for trees due to Coornaert, the existence of a boundary map by a method of Floyd, with a classical theorem of Babbage-Enriques-Petri on equations for the canonical embedding of a curve.<br />Comment: 17 pages, 4 figures
- Subjects :
- Pure mathematics
Lebesgue measure
Applied Mathematics
General Mathematics
Riemann surface
Mathematical analysis
Dynamical Systems (math.DS)
Absolute continuity
symbols.namesake
Mathematics - Algebraic Geometry
Rigidity (electromagnetism)
Poincaré conjecture
symbols
FOS: Mathematics
Embedding
Isomorphism
Mathematics - Dynamical Systems
Classical theorem
Algebraic Geometry (math.AG)
Mathematics
Subjects
Details
- Language :
- English
- ISSN :
- 01433857
- Database :
- OpenAIRE
- Journal :
- Ergodic Theory and Dynamical Systems, 33(3), 851. Cambridge University Press, Ergodic Theory and Dynamical Systems. Cambridge University Press
- Accession number :
- edsair.doi.dedup.....7dc0f0a4cce49b82a006a0dc2a417770