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 $f(z,w)=(\frac{1}{4}z+\frac{1}{2}w,w^2)$.

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 $(fN,f^{-1})$.

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.

I have no idea about the solenoid situation.

Also the topological dimension of $\cap_{n\ge1}f^nN$ is 1. I also want to know if there are higher dimensional solenoids.



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.

alt text http://dl.dropbox.com/u/5390048/LinksForDynamics.png

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.

For the Whitehead link, the limit set is a famous example known as the Whitehead continuum, 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.

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.

  • $\begingroup$ Thanks! The pictures you depicted are very clear and easy to catch. $\endgroup$ – Pengfei Jan 4 '11 at 1:44
  • $\begingroup$ By the way: we get $S^3$ if we glue two solid tori along their boundaries (like in a Hopf fiberation). What would we get if we glue two copies of $\mathbb{D}^2\times\mathbb{T}^2$ along their boundaries? $\endgroup$ – Pengfei Jan 4 '11 at 1:49
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    $\begingroup$ @Pengfei: When gluing two solid tori together, there are actually many possibilities of what you can get, because there are many different homeomorphisms $T^2 \rightarrow T^2$. You could get $S^2 \times S^1$, or you could get any lens space $L(p,q)$. For $D^2\times T^2$, there are also a lot of possibilities, depending on how you glue, including any of the above examples $\times S^1$, plus some others: determined by the pair of kernels of the epimorphisms $\mathbb Z^3 = \pi_1(T^3) \rightarrow \mathbb Z^2 = \pi_1(T^2 \times D^2$)$, up to $GL_3(\mathbb Z)$. $\endgroup$ – Bill Thurston Jan 4 '11 at 3:19

link text and link text just deal with such a topic.

  • $\begingroup$ Nice to meet you here too. But such a diff always isn't struc stable (the argument can be found in Wang's paper). It is intersting to find a closed 3 manifold with a struc stable diff such that there exists a Smale solenoid attractor. Solenoid can't be embedded into surfaces (by some theorems of Bing, reproved by Boju,Jiang et al).Some guys (Grines, Bonatti, et al) are intersted in find a stuc stable diff on 3 mfd s.t. there exists a hyp attractor which can't be embedded in surfaces. So, if you find a example I mentioned above, you give such a example $\endgroup$ – Bin Yu Dec 5 '11 at 20:49

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