Lamination (topology)


Lamination (topology)
Lamination associated with Mandelbrot set
Lamination of rabbit Julia set

In topology, a branch of mathematics, a lamination is a :

  • "A topological space partitioned into subsets"[1]
  • decoration (a structure or property at a point) of a manifold in which some subset of the manifold is partitioned into sheets of some lower dimension, and the sheets are locally parallel.

Lamination of surface is a partition of closed subset of surface into unions of smooth curves

It may or may not be possible to fill the gaps in a lamination to make a foliation.[2]

Examples

  • A geodesic lamination of a 2-dimensional hyperbolic manifold is a closed subset together with a foliation of this closed subset by geodesics[3]. These are used in Thurston's classification of elements of the mapping class group and in his theory of earthquake maps.
  • Quadratic laminations, which remain invariant under the angle doubling map.[4] These laminations are associated with quadratic maps [5][6]. It is a closed collection of chords in the unit disc [7]. It is also topological model of Mandelbrot or Julia set.

References

  1. ^ Lamination in The Online Encyclopaedia of Mathematics 2002 Springer-Verlag Berlin Heidelberg New York
  2. ^ http://www.ornl.gov/sci/ortep/topology/defs.txt Oak Ridge National Laboratory
  3. ^ Laminations and foliations in dynamics, geometry and topology: proceedings of the conference on laminations and foliations in dynamics, geometry and topology, May 18-24, 1998, SUNY at Stony Brook
  4. ^ Houghton, Jeffrey. "Useful Tools in the Study of Laminations" Paper presented at the annual meeting of the The Mathematical Association of America MathFest, Omni William Penn, Pittsburgh, PA, Aug 05, 2010
  5. ^ Tomoki KAWAHIRA: Topology of Lyubich-Minsky's laminations for quadratic maps: deformation and rigidity (3 heures)
  6. ^ Topological models for some quadratic rational maps by Vladlen Timorin
  7. ^ Modeling Julia Sets with Laminations: An Alternative Definition by Debra Mimbs

See also