Glue two solenoids along their boundaries

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!

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.

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.

• Thanks! The pictures you depicted are very clear and easy to catch. – Pengfei Jan 4 '11 at 1:44
• 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? – Pengfei Jan 4 '11 at 1:49
• @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)\$. – Bill Thurston Jan 4 '11 at 3:19