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A continuum is a nonempty compact, connected metric space.

A continuum $X$ is called arc-like if, for every $\varepsilon>0$, there is a continuous and surjective function $f:X\to [0,1]$ such that $f^{-1}(t)$ has diameter less than $\varepsilon$ for every $t\in[0,1]$.

(Equivalently, $X$ is homeomorphic to an inverse limit of arcs with surjective bonding maps.)

Arc-like continua are also called "snake-like" or "chainable" continua. For more background, see Nadler's excellent textbook 'Continuum Theory: An Introduction'.

Examples include the arc, the $\sin(1/x)$-continuum, the Knaster bucket-handle and, perhaps most famously of all, the pseudo-arc (which is the unique hereditarily indecomposable arc-like continuum).

It is easy to make a nice picture of the bucket-handle (and of the $\sin(1/x)$-continuum). As far as I know, there isn't really any good way to make a sensible picture of the pseudo-arc.

I am writing a paper that involves arc-like continua, and I would be interested to know:

Are there other interesting examples of arc-like continua that lend themselves to making nice and illuminating computer pictures?

(Of course we could combine the above examples to create new arc-like continua, but I wouldn't class this as being 'interesting'. Nadler's book has an example of a hereditarily decomposable arc-like continuum that contains no arc, but it would seem difficult to turn this into a sensible picture.)

Any pointers (or, even better, pictures!) would be appreciated. As the question is open-ended, I'm making it Community Wiki.

(In case you are interested, the main result of my paper states that there is a transcendental entire function $f:\mathbb{C}\to\mathbb{C}$ with the following property. If $X$ is an arc-like continuum with a terminal point $x_0\in X$, then there is a component $C$ of the Julia set $J(f)$ such that $C\cup\{\infty\}$ is homeomorphic to $X$. In particular, the pseudo-arc can appear as (the one-point compactification of) a component of the Julia set of a transcendental entire function.)

EDIT. The paper, Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture, is now available as preprint arXiv:1610.06278.

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  • $\begingroup$ In the parenthetical statement of your recent paper at the end of your question, I suspect there is a typo. What is the role of $X$ in the conclusion of the theorem? $\endgroup$
    – Lee Mosher
    Commented Nov 14, 2012 at 13:57
  • $\begingroup$ Oops - thanks a lot for pointing that out. Should be fixed now. :) $\endgroup$
    – Lasse Rempe
    Commented Nov 14, 2012 at 16:17
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    $\begingroup$ Just for the record (I am sure you are aware of it) there is a paper by Wayne Lewis and Piotr Minc, Drawing the pseudo-arc, Houston J. Math 2010. From their abstract: We notice that no black and white image can look hereditarily indecomposable on any raster device (like a computer screen or a printed page). We also try to generate the best computer picture of the pseudo-arc as it is possible under the circumstances. With that purpose in mind, we expand the pseudo-arc into an inverse limit with relatively simple, deterministically defined and easy to handle numerically n-crooked bonding maps. $\endgroup$
    – Mirko
    Commented Jun 3, 2023 at 1:40
  • $\begingroup$ @Mirko Thank you Mirko - yes I remember seeing that paper! $\endgroup$
    – Lasse Rempe
    Commented Jun 21, 2023 at 15:15

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Have you tried solenoids? Solenoids certainly seem like they would satisfy your definition. Also, Antoine's necklace is another likely candidate.

Edit: Sorry, I didn't realize you required it to be connected. Solenoids still work, but Antoine's necklace is out.

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    $\begingroup$ Solenoids are not arc-like. (In fact, every arc-like continuum can be embedded in the plane.) Coincidentally, Arnaud Cheritat has a beautiful picture of a solenoid on his webpage: math.univ-toulouse.fr/~cheritat/fractal/fractal.html He also has a picture of Antoine's necklace. $\endgroup$
    – Lasse Rempe
    Commented Nov 14, 2012 at 9:17

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