Beautiful examples of arc-like continua - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T21:20:25Z http://mathoverflow.net/feeds/question/112317 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112317/beautiful-examples-of-arc-like-continua Beautiful examples of arc-like continua Lasse Rempe-Gillen 2012-11-13T21:02:54Z 2012-11-14T16:17:23Z <p>A <strong>continuum</strong> is a nonempty compact, connected metric space. </p> <p>A continuum $X$ is called <strong>arc-like</strong> 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 $t\in[0,1]$. </p> <p>(Equivalently, $X$ is homeomorphic to an inverse limit of arcs with surjective bonding maps.)</p> <p>Arc-like continua are also called "snake-like" or "chainable" continua. For more background, see Nadler's excellent textbook 'Continuum Theory: An Introduction'.</p> <hr> <p>Examples include the arc, the $\sin(1/x)$-continuum, the <a href="http://commons.wikimedia.org/wiki/File%3aThe_Knaster_%22bucket-handle%22_continuum.svg" rel="nofollow">Knaster bucket-handle</a> and, perhaps most famously of all, the pseudo-arc (which is the unique hereditarily indecomposable arc-like continuum).</p> <p>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.</p> <p>I am writing a paper that involves arc-like continua, and I would be interested to know:</p> <p><strong>Are there other interesting examples of arc-like continua that lend themselves to making nice and illuminating computer pictures?</strong></p> <p>(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.)</p> <p>Any pointers (or, even better, pictures!) would be appreciated. As the question is open-ended, I'm making it Community Wiki.</p> <hr> <p>(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.)</p> http://mathoverflow.net/questions/112317/beautiful-examples-of-arc-like-continua/112343#112343 Answer by Brian Rushton for Beautiful examples of arc-like continua Brian Rushton 2012-11-14T04:12:19Z 2012-11-14T04:22:47Z <p>Have you tried solenoids? <a href="http://en.wikipedia.org/wiki/Solenoid_%2528mathematics%2529" rel="nofollow">Solenoids</a> certainly seem like they would satisfy your definition. Also, <a href="http://en.wikipedia.org/wiki/Antoine%2527s_necklace" rel="nofollow">Antoine's necklace</a> is another likely candidate.</p> <p><strong>Edit:</strong> Sorry, I didn't realize you required it to be connected. Solenoids still work, but Antoine's necklace is out.</p>