The geometry of laminations

Robbert Fokkink; Lex Oversteegen

Fundamenta Mathematicae (1996)

  • Volume: 151, Issue: 3, page 195-207
  • ISSN: 0016-2736

Abstract

top
A lamination is a continuum which locally is the product of a Cantor set and an arc. We investigate the topological structure and embedding properties of laminations. We prove that a nondegenerate lamination cannot be tree-like and that a planar lamination has at least four complementary domains. Furthermore, a lamination in the plane can be obtained by a lakes of Wada construction.

How to cite

top

Fokkink, Robbert, and Oversteegen, Lex. "The geometry of laminations." Fundamenta Mathematicae 151.3 (1996): 195-207. <http://eudml.org/doc/212192>.

@article{Fokkink1996,
abstract = {A lamination is a continuum which locally is the product of a Cantor set and an arc. We investigate the topological structure and embedding properties of laminations. We prove that a nondegenerate lamination cannot be tree-like and that a planar lamination has at least four complementary domains. Furthermore, a lamination in the plane can be obtained by a lakes of Wada construction.},
author = {Fokkink, Robbert, Oversteegen, Lex},
journal = {Fundamenta Mathematicae},
keywords = {attractor; lamination; hyperbolic geometry; tree-like continuum},
language = {eng},
number = {3},
pages = {195-207},
title = {The geometry of laminations},
url = {http://eudml.org/doc/212192},
volume = {151},
year = {1996},
}

TY - JOUR
AU - Fokkink, Robbert
AU - Oversteegen, Lex
TI - The geometry of laminations
JO - Fundamenta Mathematicae
PY - 1996
VL - 151
IS - 3
SP - 195
EP - 207
AB - A lamination is a continuum which locally is the product of a Cantor set and an arc. We investigate the topological structure and embedding properties of laminations. We prove that a nondegenerate lamination cannot be tree-like and that a planar lamination has at least four complementary domains. Furthermore, a lamination in the plane can be obtained by a lakes of Wada construction.
LA - eng
KW - attractor; lamination; hyperbolic geometry; tree-like continuum
UR - http://eudml.org/doc/212192
ER -

References

top
  1. [A-F] J. M. Aarts and R. J. Fokkink, On composants of the bucket handle, Fund. Math. 139 (1992), 193-208. Zbl0759.54017
  2. [A-H-O] J. M. Aarts, C. L. Hagopian and L. G. Oversteegen, Orientability of matchbox manifolds, Pacific J. Math. 150 (1991), 1-11. Zbl0769.54043
  3. [A-M] J. M. Aarts and M. Martens, Flows on one-dimensional spaces, Fund. Math. 131 (1988), 53-67. Zbl0677.54032
  4. [A-O] J. M. Aarts and L. G. Oversteegen, The geometry of Julia sets in the exponential family, Trans. Amer. Math. Soc. 338 (1993), 897-918. Zbl0809.54034
  5. [A-O 2] J. M. Aarts and L. G. Oversteegen, Flowbox manifolds, Trans. Amer. Math. Soc. 327 (1991), 449-463. Zbl0768.54027
  6. [B-F] H. Bohr and W. Fenchel, Ein Satz über stabile Bewegungen in der Ebene, Dansk Mat. Fys. Med. 14 (1936), 3-15. Zbl0014.11001
  7. [C-B] A. J. Casson and S. A. Bleiler, Automorphisms of Surfaces after Nielsen and Thurston, Cambridge Univ. Press, 1988. Zbl0649.57008
  8. [C-C] J. H. Case and R. E. Chamberlain, Characterization of tree-like continua, Pacific J. Math. 10 (1960), 73-84. 
  9. [C-N] C. Camacho and A. L. Neto, Geometric Theory of Foliations, Birkhäuser, Basel, 1985. 
  10. [D] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Redwood City, 1987. 
  11. [F] R. J. Fokkink, The structure of trajectories, Thesis, University of Delft, 1991. 
  12. [G] C. Gutierrez, Structural stability for flows on the torus with a cross-cap, Trans. Amer. Math. Soc. 241 (1978), 311-320. Zbl0396.58017
  13. [Ka] H. Kato, On expansiveness of shift homeomorphisms of inverse limits of graphs, Fund. Math. 137 (1991), 201-210. Zbl0738.54017
  14. [Kn] B. Knaster, Fund. Math. 3 (1922), 209. 
  15. [K] N. N. Konstantinov, Complement of selfcoiling curves in the plane, Fund. Math. 57 (1965), 147-171. 
  16. [K2] N. N. Konstantinov, Non-self-intersecting curves in the plane, Mat. Sb. (N.S.) 54 (1961), 253-294 (in Russian). 
  17. [K-S] H. B. Keynes and M. Sears, Modeling expansion in real flows, Pacific J. Math. 85 (1979), 111-124. 
  18. [M-P] W. de Melo and J. Palis Jr., Geometric Theory of Dynamical Systems. An Introduction, Springer, New York, 1982. 
  19. [P] M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962), 101. Zbl0107.07103
  20. [Pl1] R. V. Plykin, Sources and sinks of A-diffeomorphisms of surfaces, Mat. Sb. 94 (1974), 243-264 (in Russian). 
  21. [Pl2] R. V. Plykin, On the geometry of hyperbolic attractors of smooth cascades, Uspekhi Mat. Nauk 39 (6) (1984), 75-113 (in Russian). 
  22. [S] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. Zbl0202.55202
  23. [T] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988), 418-431. Zbl0674.57008
  24. [Wh] H. Whitney, Regular families of curves, Ann. of Math. 34 (1933), 244-270. Zbl0006.37101
  25. [Wi1] R. F. Williams, Expanding attractors, Inst. Hautes Etudes Sci. Publ. Math. 43 (1974), 169-203. 
  26. [Wi2] R. F. Williams, The DA map of Smale and structural stability, in: Global Analysis, Proc. Sympos. Pure Math. 14, 1970, 329-334. 
  27. [Z1] A.Yu. Zhirov, A list of hyperbolic attractors on orientable surfaces and applications to pseudo-Anosov homeomorphisms, Dokl. Akad. Nauk 330 (1993), 683-686 (in Russian). 
  28. [Z2] A.Yu. Zhirov, Hyperbolic attractors of diffeomorphisms of orientable surfaces, Mat. Sb. 186 (1995), 59-82 (in Russian). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.