Glue two solenoids along their boundaries - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T21:40:24Z http://mathoverflow.net/feeds/question/51018 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51018/glue-two-solenoids-along-their-boundaries Glue two solenoids along their boundaries Pengfei 2011-01-03T14:28:12Z 2011-11-08T04:04:51Z <p>Here a solenoid is a dynamical system $(N,f)$ where $N$ is the solid torus $N=\mathbb{D}^2\times S^1$ with boundary $S^1\times S^1$, and $f:N\to N$ is a smooth embedding whose image is wrapped twice in $N$. For example Smale solenoid $f(z,w)=(\frac{1}{4}z+\frac{1}{2}w,w^2)$.</p> <p>I am wondering if we can glue two solenoids together to formulate a diffeomorphism. What I have in mind is consider the diffeomorphism $f:N\to fN$ and $f^{-1}:fN\to N$. I want to glue the two disjoint copies $(N,f)$ and $(fN,f^{-1})$.</p> <p>The basic picture for it is to glue $g:\mathbb{D}^2\to \mathbb{D}^2,x\mapsto x/2$ with $g^{-1}:g\mathbb{D}^2\to \mathbb{D}^2$. We can add a collary to their boundaries on which $g$ and $g^{-1}$ can be glued. The result manifold is just the two-sphere $S^2$ and the map is the North--South map.</p> <p>I have no idea about the solenoid situation. </p> <p>Also the topological dimension of $\cap_{n\ge1}f^nN$ is 1. I also want to know if there are higher dimensional solenoids. </p> <p>Thanks! </p> http://mathoverflow.net/questions/51018/glue-two-solenoids-along-their-boundaries/51042#51042 Answer by Bill Thurston for Glue two solenoids along their boundaries Bill Thurston 2011-01-03T17:58:25Z 2011-01-03T17:58:25Z <p>Yes, there are diffeomorphisms of $S^3$ as you suggest. Here's one slightly more general construction: start from a regular neighborhood of any link in $S^3$ made from two unknotted circles. Two examples are shown below, the first associated with your example is the $(4,2)$-torus link, the second is the Whitehead link. Since the circles are unknotted, the complement of each solid torus is also a solid torus, and there is a diffeomorphism of $S^3$ sending one of the tori to the other; it sends the first solid torus to the complement of the second solid torus, and the complement of the first solid torus to the second solid torus.</p> <p><img src="http://dl.dropbox.com/u/5390048/LinksForDynamics.png" alt="alt text"></p> <p>For the torus link example on the left, the diffeomorphism can be chosen so that the forward limit set of almost every point is a solenoid inside the second solid torus, and the backward limit set is a solenoid inside the other.</p> <p>For the Whitehead link, the limit set is a famous example known as the <a href="http://en.wikipedia.org/wiki/Whitehead_manifold" rel="nofollow">Whitehead continuum</a>, whose complement in $S^3$ is a simply-connected non-compact 3-manifold. Whitehead constructed it to demolish a proof he thought he had found for the Poincaré conjecture.</p> <p>And yes, higher dimensional examples can be constructed in much the same way: for example, to get an example with topological dimension $n$, you can start from an expanding self-covering map $T^n \rightarrow T^n$, then lift this to an embedding of $T^n \times D^2$ into $T^n \times D^2$ that is a contraction in the $D^2$ direction. The limit set is a solenoid. There are many variations of these constructions, and limit sets can become quite complicated.</p> http://mathoverflow.net/questions/51018/glue-two-solenoids-along-their-boundaries/80356#80356 Answer by Bin Yu for Glue two solenoids along their boundaries Bin Yu 2011-11-08T04:04:51Z 2011-11-08T04:04:51Z <p><a href="http://3-manifolds%20that%20admit%20knotted%20solenoids%20as%20attractors/" rel="nofollow">link text</a> and <a href="http://www.sciencedirect.com/science/article/pii/S0166864107002258" rel="nofollow">link text</a> just deal with such a topic.</p>