Talk:Solenoid (mathematics)

Is this correct?

The article describes its construction of an embedding of the p-adic solenoid in R³ as follows:

"Take a solid torus T in R³ and choose an embedding α: T → T such that α acts on the fundamental group of T as multiplication by p; that is to say, α maps T onto a solid torus inside T which winds p times around the axis of T before joining up with itself. Then . . . [t]he intersection (in R³) of the smaller and smaller toruses T, αT, α(αT), etc., is a p-adic solenoid inside T, hence in R³."

But I am unconvinced that it is sufficient to express the embedding of one solid torus in the next solely in terms of homotopy. (E.g., what if a knot were introduced into the image of α without changing its homotopy class???)

Can anyone point to a proof that the quoted conditions on α are actually sufficient?

It is safe to say, however, that if -- in addition to the homotopy condition -- the embedding α carries each 2-disk fibre of the domain T = D² x S¹ (that is, each D² x {p}) into the interior of a 2-disk fibre D² x {q} of the codomain T = D² x S¹, then we can be sure that the inverse limit is indeed homeomorphic to the intended solenoid.Daqu (talk) 21:51, 30 September 2008 (UTC)

This is not correct, either!

The article presents the following as the geometric construction of a solenoid:

"Each solenoid may be constructed as the intersection of a nested system of embedded solid tori in R³.

"Fix a sequence of natural numbers {n_i}, n_i ≥ 2. Let T₀ = S¹ × D be a solid torus. For each i ≥ 0, choose a solid torus T_i+1 that is wrapped longitudinally n_i times inside the solid torus T_i. Then their intersection . . . is homeomorphic to the solenoid constructed as the inverse limit of the system of circles with the maps determined by the sequence {n_i}."

But this construction does not give enough detail to ensure that the resulting intersection is in fact "homeomorphic to the inverse limit of the system of circles with the maps determined by the sequence {n_i}", or to ensure that it is in fact a solenoid at all. For example, there is nothing to prevent the intersection of the T_i, i=0,1,2,3,..., from having an interior point in R³.Daqu (talk) 14:52, 4 February 2012 (UTC)

If we define the "thickness" of T_i to be the maximum of d(x,R³ \ T_i) over all x in $T_i then as long as the thickness goes to 0 as i goes to infinity, then we do get the solenoid. Is there a way to say this without defining something new? (Has "thickness" already been defined, perhaps under a different name?) 92.201.209.63 (talk) 12:51, 24 October 2012 (UTC)

Certain terms can be ambiguous / have no reference

The page starts with:

In mathematics, a solenoid is a compact connected topological space (i.e. a continuum) that may be obtained as the inverse limit of an inverse system of topological groups and continuous homomorphisms

$f_{i}:S_{i+1}\to S_{i}\quad \forall i\geq 0$

where each $S_{i}$ is a circle and f_i is the map that uniformly wraps the circle $S_{i+1}$ for $n_{i+1}$ times ($n_{i+1}\geq 2$) around the circle $S_{i}$.

However what is a circle in this context? Is it $S^{1}$, is it just a set of points with no topology? And what does "uniformly wraps" mean in this context? Someone that's not familiar with the topic, but can be familiar with the necessary mathematics to understand it, may not be able to understand the definition. Average complainer (talk) 20:07, 22 June 2023 (UTC)

Group structure

There are almost no details about the group structure of the solenoid.

It would be a worthy addition to this article if someone knowledgeable about the subject introduced a section showing a coordinate system for (at least) the dyadic solenoid, and how to calculate a) addition and b) inversion, in those coordinates.

This can't be right

The section about symbolic dynamics contains this clause:

"Viewed as a set, the solenoid is just a Cantor-continuum of circles".

But the circle cannot embed topologically in the solenoid.

So there is something seriously wrong with this statement.

I hope it can be corrected by someone knowledgeable about the subject.