Talk:Horseshoe map

Talk 2004
(William M. Connolley 16:42, 2004 Mar 26 (UTC)) Umm, is the page correct? If x>0.5, 1-2x<0 and the point leaves the unit square. Or have I missed something? Quite likely...

The second-to-last paragraph seems redundant; at least the beginning of it does. What do you think? Cluster 01:47, 6 May 2004 (UTC)

To do:
 * explain the formation of the invariant set (XaosBits 04:52 GMT, 4 Apr 2005)
 * relate horseshoe maps to flows and homoclinic tangles
 * explain pruning and how it relates to Thurston's classification

Baker x Horseshoe
This page describes what is known as the Baker map. The Smale horseshoe compresses a square into a horseshoe shaped region. I suggest that the content be moved to a new Baker_map page. Moved (XaosBits). The page now described the Smale horseshoe.


 * Horseshoe map is NOT Baker map. For example, one can extend horseshoe map to the diffeomorphism of sphere, that cannot be done with Baker map, because of it's nonsmoothness. -- 85.140.10.27 19:05, 19 Mar 2005 (UTC) (Ilya Schurov).

optical universal Turing machine
"Using a few hundred mirrors, one can build an optical universal Turing machine in one's backyard, using the horseshoe map."

Please, could you explain more precise what do you mean and where one can find more information on this topic?

-- 83.237.103.131 21:18, 20 Mar 2005 (UTC) (Ilya Schurov)

Quote from Turing machine:


 * Also, using a few hundred mirrors, one can build an optical universal Turing machine in one's backyard, using the Horseshoe map. This is based on a work by Stephen Smale.

-- Sy / (talk)

The comment about Turing machines, horseshoe maps, and mirrors, I think, is related to the observations that Chris Moore made in his article Unpredictability and undecidability in dynamical systems in Phys. Rev. Lett. 64:2354–2357 (May 1990). There Chris maps a Turing machine into a generalized shift map and then observes that these shift maps correspond to the symbolic dynamics of billiards. I am moving this to the Baker map page until it can be further edited. -- XaosBits, Wed Mar 23 04:53 GMT 2005

formula
This article should give the formula for this map. --MarSch 13:53, 2 Jun 2005 (UTC)


 * The horseshoe map is defined geometrically and is really a class of maps.  The actual formula for any one of them does not matter much and is quite messy.  If there is interest, I could paste the code I used for the illustrations (which contains a definition of one member from the class of horseshoe maps).   XaosBits 17:31, 2 Jun 2005 (UTC)

Embeddings?
Something about the embedding of the horseshoe into other systems (in particlar into the restricted three-body problem) would help explain why it's important. —The preceding unsigned comment was added by 199.17.27.38 (talk) 21:13, 31 January 2007 (UTC).

What is the domain?
The article begins by saying that a horseshoe map is a map of a square into itself, but illustrations immediately below use an oval (not even a rectangle). On the other hand, the later, more technical section operates with a region of the plane of a fairly special kind and, while consistent with the illustration, it doesn't match the usual conventions about what constitutes a Smale's horseshoe. All this needs to be explained better. Arcfrk 02:01, 21 September 2007 (UTC)

Consistency of Description and Illustration
Thanks Arcfrk. I have similar concerns. The description has "... defined geometrically by squishing the square, ..." whereas the diagram shows an oblong. Is this map applied to a unit square in R2 or to a unit square slab in R3? The "square" in the description refers to the plan view of R2? The oblong in the illustration represents the side view of a batch of dough? The description and illustration should be consistent. If the intention is to illustrate an object in three dimensions, an isometric view or orthographic projection should be used; or least the description should mention which view the illustration portrays. Regards, PeterEasthope (talk) 15:33, 16 December 2011 (UTC)

If we're talking about the 'iterating the square' illustration, I'm with you. First, the square itself should show no end caps on the forward map. And then the inverse map looks like a chopped off reflection, when it should be vertical bars at each side of the square, snug to the sides. I have no idea what's meant by the parts sticking up above the square in the inverses, which is outside the space the square or caps ever map to.

173.25.54.191 (talk) 01:52, 29 December 2012 (UTC)

Curiosity: pastry
It is curious that the variant map in its curled up form is that used to make puff pastry! I fear, however, that it is probably not really worth mentioning in the article — nor indeed worth mentioning this map in that article. PJTraill (talk) 17:37, 24 August 2014 (UTC)