Kontsevich–Zorich cocycle
E904569
The Kontsevich–Zorich cocycle is a dynamical system arising from the action of the Teichmüller geodesic flow on the Hodge bundle over moduli spaces of Riemann surfaces, central to understanding deviations of ergodic averages and Lyapunov exponents in flat surface dynamics.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
cocycle
ⓘ
dynamical system ⓘ mathematical object ⓘ |
| actsOn |
Hodge bundle over Teichmüller space
ⓘ
cohomology of Riemann surfaces ⓘ first homology group ⓘ |
| appearsIn |
study of Abelian differentials on Riemann surfaces
ⓘ
study of billiards in rational polygons ⓘ study of measured foliations on surfaces ⓘ |
| arisesFrom | Teichmüller geodesic flow NERFINISHED ⓘ |
| associatedWith |
Gauss–Manin connection
NERFINISHED
ⓘ
Hodge theory NERFINISHED ⓘ SL(2,R)-action on moduli spaces ⓘ Teichmüller flow NERFINISHED ⓘ differential geometry ⓘ dynamical systems ⓘ ergodic theory ⓘ |
| definedAs | cocycle given by parallel transport of cohomology along Teichmüller geodesic flow ⓘ |
| definedOn | Hodge bundle NERFINISHED ⓘ |
| definedOver |
moduli space of Abelian differentials
ⓘ
moduli space of Riemann surfaces ⓘ moduli space of quadratic differentials NERFINISHED ⓘ |
| hasComponent |
central Oseledets subspace
ⓘ
stable Oseledets subspace ⓘ unstable Oseledets subspace ⓘ |
| hasInvariant |
Kontsevich–Zorich Lyapunov exponents
NERFINISHED
ⓘ
Lyapunov spectrum ⓘ |
| hasProperty |
linear
ⓘ
measurable ⓘ symplectic ⓘ |
| introducedBy |
Anton Zorich
NERFINISHED
ⓘ
Maxim Kontsevich NERFINISHED ⓘ |
| namedAfter |
Anton Zorich
NERFINISHED
ⓘ
Maxim Kontsevich NERFINISHED ⓘ |
| relatedTo |
Eskin–Kontsevich–Zorich formula for Lyapunov exponents
ⓘ
Forni subspace NERFINISHED ⓘ Forni’s deviation theorem NERFINISHED ⓘ Oseledets multiplicative ergodic theorem NERFINISHED ⓘ flat surface dynamics ⓘ interval exchange transformations ⓘ translation surfaces ⓘ |
| studiedIn |
Teichmüller dynamics
NERFINISHED
ⓘ
flat geometry ⓘ moduli of curves ⓘ |
| usedFor |
study of Lyapunov exponents
ⓘ
study of deviation of Birkhoff sums ⓘ study of deviation of ergodic integrals ⓘ study of deviations of ergodic averages ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.