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By a Knaster continuum is meant a continuum homeomorphic to the inverse limit of a sequence of unit intervals with open, non-homeomorphic bonding maps . Denote by the class of all Knaster continua.

The most popular examples of Knaster continua is the buckethandle (sometimes called by dynamists the horseshoe ) , where , and the double buckethandle with

Nice geometric models of and can be described in the following way (see [Kuratowski 1968, pp. 204-205]). If denotes the standard Cantor ternary set in , then is homeomorphic to the union of all semi-circles

where , and all semi-circles

for , where .

See Figure A.

If is the quintary Cantor set in (real numbers in which can be written in the enumeration system at base 5 without digits 1 and 3), then is homeomorphic to the union of all semi-circles

for and and all semi-circles

for and .

See Figure B.

1. If is the map such that for even , for odd , where is an integer between 0 and , and is linear on each interval , then is equal to the family of all continua homeomorphic to inverse limits , where, for each , and for some [Rogers 1970]. So, each is determined by a sequence such that . We will denote such by and call it a -adic Knaster continuum . Without loss of generality, one can assume that all 's are prime.

2. Each -adic Knaster continuum is homeomorphic to the quotient of the -adic solenoid under the relation

or

where and is the group inverse of . A point of is an end point of if and only if it is an image under the quotient map of the neutral element or the element of order 2 (if such exists) of the corresponding solenoid (see, e.g., [Krupski 1984b, p. 43]).

3. The class of all Knaster continua is equal to the class of all arc-like continua with one or two end points having the property of Kelley and arcs as proper non-degenerate subcontinua and which themselves are not arcs [Krupski 1984b]. In particular, all of them are indecomposable.

4. There are mutually non-homeomorphic members of [Wa]; in fact, there is a subfamily of of cardinality no member of which is an open image of another one [Debski 1985].

5. Any open map between two Knaster continua , is the uniform limit of open maps which are induced by maps between the inverse sequences [Eberhart et al. 1999, Theorem 4.8, p. 145].

6. If , then every indecomposable continuum can be mapped onto [Rogers 1970, Corollary, p. 455].

Moreover, if is an indecomposable (Hausdorff) continuum, and ( and belong to different composants of ), then there is a continuous surjection such that ( and belong to different composants of , resp.). It follows that any indecomposable continuum embeds in the Cartesian product of copies of [Bellamy 1973, Corollaries 1 and 2, p. 305].

7. If then there are distinct homotopy types of maps of onto that map an end point of onto an end point of [Minc 1999].

8. If and is a monotone map of , then is homeomorphic to [Krupski 1984b].

9. There is no exactly 2-to-1 map defined on a Knaster continuum [Debski 1992].
10. Knaster continua are absolute retracts for the class of all hereditarily unicoherent continua, i.e., if , is a hereditarily unicoherent continuum and , then is a retract of . Equivalently, if is a closed subset of a hereditarily unicoherent continuum , then every map from onto a Knaster continuum can be extended over ([Mackowiak 1984] for and a Hausdorff hereditarily unicoherent continuum and [Charatonik et al. XXXXa] for arbitrary ).

11. Any two points of a plane continuum can be joined by a hereditarily decomposable subcontinuum if and only if cannot be mapped onto [Hagopian 1974, Theorem 2, p. 133].

12. Every autohomeomorphism of the buckethandle is isotopic to some iterate of the shift homeomorphism [Aarts et al. 1991, Theorem 4.4, p. 204].

Moreover, any autohomeomorphism of a Knaster continuum is isotopic to a standard homeomorphism (defined as a map induced by a map of inverse systems); maps and have the same topological entropy and if the entropy is positive, then they are semi-conjugate [Kwapisz 2001, Theorem 4, p. 271].

13. If is the end point of and are composants of which do not contain , then there exists a continuous injection such that [Aarts et al. 1991, Theorem 5.1].

Any two composants of which do not contain are homeomorphic [Bandt 1994].

14. Every autohomeomorphism of the buckethandle continuum has at least two fixed points [Aarts et al. 1998].

15. If is a Borel subset of and is the union of a family of composants of , then is meager or comeager [Emeryk 1980], [Krasinkiewicz 1974a].

16. Each composant of different from the one containing the end point is internal, i.e., every continuum intersecting both and must intersect all composants of [Krasinkiewicz 1974b, Theorem, p. 261].

17. Any Knaster continuum , where for each , is homeomorphic to the attracting set of a homeomorphism of the plane closed disk [Barge 1986]. If for all (i.e., ), then can be defined as a diffeomorphism which is called a horse-shoe map--it was first described by S. Smale in [Smale 1965] (see also [Barge 1988]). See Figure C.

18. The buckethandle and all Knaster continua with two end points admit no mean [Illanes XXXX], [Kawamura et al. 1996, Theorem 2.2, p. 99].

19. The hyperspace of all subcontinua of any Knaster continuum is homeomorphic to the cone over by a homeomorphism which sends onto the vertex of and the set of all singletons of onto the base of the cone [Dilks et al. 1981, Corollary 12, p. 639].

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Next: Unicoherent continua (also hereditary) Up: Indecomposable continua (also hereditary) Previous: Indecomposable continua (also hereditary)
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30