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

• Conformal Laminations Thesis by Vineet Gupta, California Institute of Technology Pasadena, California 2004

See also

• Train track (topology)
• Orbit portrait

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