Veech surface
E904568
A Veech surface is a special type of translation surface whose affine symmetry group is a lattice in SL(2,ℝ), giving it particularly rigid and highly structured dynamical and geometric properties.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
dynamical system
ⓘ
mathematical object ⓘ translation surface ⓘ |
| example |
L-shaped translation surfaces of certain parameters
ⓘ
regular n-gon translation surfaces for certain n ⓘ regular octagon translation surface ⓘ |
| generalizationOf | square-tiled surface in some cases ⓘ |
| hasInvariant | Veech group NERFINISHED ⓘ |
| hasProperty |
Teichmüller curve of finite volume in moduli space
ⓘ
Teichmüller disk with lattice stabilizer ⓘ Veech dichotomy NERFINISHED ⓘ Veech group acts discretely on the upper half-plane ⓘ affine automorphism group of finite covolume in SL(2,R) ⓘ affine automorphisms preserve the flat structure ⓘ affine symmetry group is a lattice in SL(2,R) ⓘ associated Teichmüller curve has finite hyperbolic area ⓘ can be viewed as an Abelian differential on a Riemann surface ⓘ closed SL(2,R)-orbit in moduli space of translation surfaces ⓘ completely periodic directions ⓘ directions with cylinder decompositions form a dense subset of S^1 ⓘ finite area ⓘ flat metric with conical singularities ⓘ geodesic flow given by straight-line flow in charts ⓘ highly structured geometry ⓘ holomorphic 1-form defines the translation structure ⓘ lattice surface ⓘ locally modeled on the Euclidean plane ⓘ optimal deviation of ergodic averages ⓘ optimal dynamics for straight-line flow ⓘ parabolic elements in affine group corresponding to cylinder decompositions ⓘ rigid dynamical behavior ⓘ straight-line flow is either completely periodic or uniquely ergodic in any given direction ⓘ unique ergodicity in almost every direction ⓘ |
| namedAfter | William A. Veech NERFINISHED ⓘ |
| relatedTo |
SL(2,R)-action on moduli spaces
ⓘ
Teichmüller curve NERFINISHED ⓘ billiard flow in rational polygons ⓘ interval exchange transformation ⓘ moduli space of Riemann surfaces ⓘ |
| studiedIn |
Teichmüller dynamics
ⓘ
billiards in polygons ⓘ ergodic theory ⓘ flat geometry ⓘ |
| usedFor |
constructing examples with extremal dynamical properties
ⓘ
studying deviation of ergodic averages for flows on flat surfaces ⓘ |
| VeechGroup | lattice subgroup of SL(2,R) ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.