Cantor tree surface

In dynamical systems, the Cantor tree is an infinite-genus surface homeomorphic to a sphere with a Cantor set removed. The blooming Cantor tree is a Cantor tree with an infinite number of handles added in such a way that every end is a limit of handles.^[1][2]

The bark of a fractal tree, splitting in two directions at each branch point, forms a Cantor tree surface. Drilling a hole through the tree at each branch point would produce a blooming Cantor tree.

An Alexander horned sphere. Its non-singular points form a Cantor tree surface.

See also

• Jacob's ladder surface
• Loch Ness monster surface

References

1. Ghys, Étienne (1995), "Topologie des feuilles génériques", Annals of Mathematics, Second Series (in French), 141 (2): 387–422, doi:10.2307/2118526, ISSN 0003-486X, JSTOR 2118526, MR 1324140
2. Walczak, Paweł (2004), Dynamics of foliations, groups and pseudogroups, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 64, Birkhäuser Verlag, p. 210, ISBN 978-3-7643-7091-6, MR 2056374