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Knaster continua

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By a Knaster continuum is meant a continuum homeomorphic to the inverse limit \varprojlim(I_n,f_n)$ of a sequence of unit intervals I_n=I$ with open, non-homeomorphic bonding maps f_n$. Denote by \mathcal K$ the class of all Knaster continua.

The most popular examples of Knaster continua is the buckethandle (sometimes called by dynamists the horseshoe ) B_1$, where f_n(t)=1-\vert 2t-1\vert$, and the double buckethandle B_2$ with

\displaystyle f_n(t)=\begin{cases}
3t& \text{if \ $0\le t\le\frac13$},\\
2-3t& ...
...frac13\le t\le \frac23$},\\
3t-2& \text{if \ $\frac23\le t\le 1$}.

Nice geometric models of B_1$ and B_2$ can be described in the following way (see [Kuratowski 1968, pp. 204-205]). If C$ denotes the standard Cantor ternary set in I$, then B_1$ is homeomorphic to the union of all semi-circles

\displaystyle \{\,(x,y)\in\mathbb{R}2: (x-\frac12)^2+y^2=r^2, y\ge 0\,\},$

where r+\frac12\in C$, and all semi-circles

\displaystyle \{\,(x,y): (x-\frac5{2\cdot3^n})^2+y^2=r^2, y\le 0\,\},$

for n \in \mathbb{N}$, where r+\frac5{2\cdot3^n}\in C\cap [\frac2{3^n},\frac1{3^{n-1}}]$.

See Figure A.

Figure 3.2.1: ( A ) buckethandle

If E$ is the quintary Cantor set in I$ (real numbers in I$ which can be written in the enumeration system at base 5 without digits 1 and 3), then B_2$ is homeomorphic to the union of all semi-circles

\displaystyle \{\,(x,y): (x-\frac7{10\cdot5^n})^2+y^2=r^2, y\le0\,\},$

for n \in \mathbb{N}$ and r+\frac7{10\cdot5^n}\in E\cap
[\frac2{5^{n+1}},\frac1{5^n}]$ and all semi-circles

\displaystyle \{\,(x,y):(x-(1-\frac7{10\cdot5^n}))^2+y^2=r^2, y\ge 0\,\},$

for n\ge 0$ and \frac2{5^{n+1}}\le\frac7{10\cdot5^n}-r\le\frac1{5^n}$.

See Figure B.

Figure 3.2.1: ( B ) double buckethandle

Figure: ( BB ) double buckethandle

  1. If w_p:I\to I$ is the map such that w_p(i/p)=0$ for even i$, w_k(i/p)=1$ for odd i$, where i$ is an integer between 0 and p\in\mathbb{N}$, and w_p$ is linear on each interval [i/p,(i+1)/p]$, then \mathcal K$ is equal to the family of all continua homeomorphic to inverse limits \varprojlim(I_n,f_n)$, where, for each n$, I_n=I$ and f_n=w_{p_n}$ for some p_n$ [Rogers 1970]. So, each K\in\mathcal K$ is determined by a sequence \mathbf{p}
=(p_1,p_2,\dots)$ such that K= \varprojlim(I_n,w_{p_n})$. We will denote such K$ by K(\mathbf{p})$ and call it a \mathbf p$-adic Knaster continuum . Without loss of generality, one can assume that all p_n$'s are prime.

  2. Each \mathbf p$-adic Knaster continuum K$ is homeomorphic to the quotient of the \mathbf{p}$-adic solenoid \Sigma(\mathbf{p})$ under the relation

    \displaystyle a\sim b\Leftrightarrow a=b$   or\displaystyle \quad a=b^{-1}, $

    where a,b\in \Sigma(\mathbf{p})$ and b^{-1}$ is the group inverse of b$. A point of K$ is an end point of K$ 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 \mathcal K$ 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 2^{\aleph_0}$ mutually non-homeomorphic members of \mathcal K$ [Wa]; in fact, there is a subfamily of \mathcal K$ of cardinality 2^{\aleph_0}$ no member of which is an open image of another one [Debski 1985].

  5. Any open map between two Knaster continua \varprojlim(I_n,f_n)$, \varprojlim(I_n,g_n)$ is the uniform limit of open maps which are induced by maps (I_n,f_n)\to (I_n,g_n)$ between the inverse sequences [Eberhart et al. 1999, Theorem 4.8, p. 145].

  6. If K\in\mathcal K$, then every indecomposable continuum can be mapped onto K$ [Rogers 1970, Corollary, p. 455].

    Moreover, if X$ is an indecomposable (Hausdorff) continuum, a,b\in X$ and a\ne b$ (a$ and b$ belong to different composants of X$), then there is a continuous surjection f:X\to B_1$ such that f(a)\ne f(b)$ (f(a)$ and f(b)$ belong to different composants of B_1$, resp.). It follows that any indecomposable continuum embeds in the Cartesian product of copies of B_1$ [Bellamy 1973, Corollaries 1 and 2, p. 305].

  7. If K_1, K_2\in \mathcal K$ then there are 2^{\aleph_0}$ distinct homotopy types of maps of K_1$ onto K_2$ that map an end point of K_1$ onto an end point of K_2$ [Minc 1999].

  8. If K\in\mathcal K$ and f$ is a monotone map of K$, then f(K)$ is homeomorphic to K$ [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 K\in\mathcal K$, X$ is a hereditarily unicoherent continuum and K\subset X$, then K$ is a retract of X$. Equivalently, if A$ is a closed subset of a hereditarily unicoherent continuum X$, then every map f$ from A$ onto a Knaster continuum can be extended over X$ ([Mackowiak 1984] for K=B_1$ and X$ a Hausdorff hereditarily unicoherent continuum and [Charatonik et al. XXXXa] for arbitrary K\in\mathcal K$).

  11. Any two points of a plane continuum X$ can be joined by a hereditarily decomposable subcontinuum if and only if X$ cannot be mapped onto B_1$ [Hagopian 1974, Theorem 2, p. 133].

  12. Every autohomeomorphism of the buckethandle B_1$ is isotopic to some iterate of the shift homeomorphism f:B_1\to B_1,
f(x_1,x_2,\dots)=(x_2,x_3,\dots)$ [Aarts et al. 1991, Theorem 4.4, p. 204].

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

  13. If e$ is the end point of B_1$ and S, T$ are composants of B_1$ which do not contain e$, then there exists a continuous injection h:B_1\setminus\{e\}\to
B_1\setminus\{e\}$ such that h(S)=T$ [Aarts et al. 1991, Theorem 5.1].

    Any two composants of B_1$ which do not contain e$ are homeomorphic [Bandt 1994].

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

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

  16. Each composant C$ of B_1$ different from the one containing the end point is internal, i.e., every continuum L\subset \mathbb{R}^2$ intersecting both C$ and \mathbb{R}^2\setminus B_1$ must intersect all composants of B_1$ [Krasinkiewicz 1974b, Theorem, p. 261].

  17. Any Knaster continuum K=\varprojlim(I_n,f_n)$, where f_m=f_n$ for each m,n$, is homeomorphic to the attracting set of a homeomorphism h:D^2\to
h(D^2)\subset D^2$ of the plane closed disk [Barge 1986]. If f_n(t)=1-\vert 2t-1\vert$ for all n$ (i.e., K=B_1$), then h$ 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.

    Figure 3.2.1: ( C ) the horse-shoe map f$ maps the stadium shaped region D$ ( A \cup R \cup B$) in such a way, that it shrinks R$ vertically, stretches R$ horizontally, contracts the semicircles then folds the space once and places this result into itself so that f(A)$ and f(B)$ are in the interior of A$ and f(R)$ is in the interior of D$.

  18. The buckethandle B_1$ 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 C(K)$ of all subcontinua of any Knaster continuum K$ is homeomorphic to the cone Cone(K)$ over K$ by a homeomorphism h:C(K)\to Cone(K)$ which sends K$ onto the vertex of Cone(K)$ and the set of all singletons of K$ onto the base of the cone [Dilks et al. 1981, Corollary 12, p. 639].

Here you can find source files of this example.

Here you can check the table of properties of individual continua.

Here you can read Notes or write to Notes ies of individual continua.
next up previous contents index
Next: Unicoherent continua (also hereditary) Up: Indecomposable continua (also hereditary) Previous: Indecomposable continua (also hereditary)
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih