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

This topology-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Topology Topology stubs Wikimedia Foundation. 2010. Игры ⚽ Нужно решить контрольную? Measured Lies: The Bell Curve Examined Nautical measured mile Look at other dictionaries: List of geometric topology topics — This is a list of geometric topology topics, by Wikipedia page. See also: topology glossary List of topology topics List of general topology topics List of algebraic topology topics Publications in topology Contents 1 Low dimensional topology 1.1 …   Wikipedia Bass–Serre theory — is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees. The theory relates group actions on trees with decomposing groups as iterated… …   Wikipedia 3-manifold — In mathematics, a 3 manifold is a 3 dimensional manifold. The topological, piecewise linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3… …   Wikipedia Rips machine — In geometric group theory, the Rips machine is a method of studying the action of groups on R trees. It was introduced in unpublished work of Eliyahu Rips in about 1991. An R tree is a uniquely arcwise connected metric space in which every arc is …   Wikipedia Train track — In the mathematical area of topology, a train track is a family of curves embedded on a surface, meeting the following conditions: #The curves meet at a finite set of vertices called switches . #Away from the switches, the curves are smooth and… …   Wikipedia Cyclic order — In mathematics, a cyclic order is a way to arrange a set of objects in a circle.[nb] Unlike most structures in order theory, a cyclic order cannot be modeled as a binary relation a < b . One does not say that east is more clockwise than west.… …   Wikipedia Allen Hatcher — Allen Edward Hatcher is an American topologist and also a noted author. His book Algebraic Topology , which is the first in a series, is considered by many to be one of the best introductions to the subject.He received his Ph.D. under the… …   Wikipedia David Gabai — Nationality  United States Fields …   Wikipedia Tameness theorem — In mathematics, the tameness theorem states that every complete hyperbolic 3 manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3 manifold. The tameness theorem was… …   Wikipedia Density theorem for Kleinian groups — In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups.… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”