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. 


Thanks! 



Glue two solenoids along their boundaries
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](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](http://en.wikipedia.org/wiki/Whitehead_manifold), 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.


