How can I tell if two functions are conjugates in the homeomorphism group of $\mathbb{S}^n$?

Question

Suppose we have two functions $f,g:\mathbb{S}^n\to\mathbb{S}^n$ which are bijective, continuous, and have a continuous inverse (aka bicontinuous). They are conjugates in the homeomorphism group when there's another bicontinuous function $c:\mathbb{S}^n\to\mathbb{S}^n$ such that

$$f\circ c = c\circ g.$$

I'm mainly interested in results that provide sufficient conditions to know when $f$ and $g$ allow this equation to be true (ie when such a $c$ exists).

I'm also curious (not the main question) if there's a classification of all conjugacy classes for this group.

Answer 1

I don't see the need to introduce the term bicontinuous function when we already have homeomorphism. Your question is about classifying homeomorphisms up to topological conjugacy. Let's look at $n=1$ first.

Definition. A homeomorphism $f: X\to X$ is transitive if for any two nonempty open sets $U,V\subset X$ there is a positive integer $n$ such that $f^n(U)\cap V\ne\varnothing$.

Theorem (Poincaré). Any transitive homeomorphism $f:S^1\to S^1$ is topologically conjugate to rotation of $S^1$ by an irrational multiple of $\pi$.

Since the rotation number is invariant under topological conjugacy, Poincaré's theorem gives a complete classification of transitive circle homeomorphisms: they are topologically conjugate if and only if they have the same rotation number.

The situation is more complicated for non-transitive homeomorphisms with an irrational rotation number. However, Denjoy proved that a $C^1$-diffeomorphism with derivative of bounded variation is topologically conjugate to irrational rotation whenever it has the irrational .

When the rotation number is rational, topological conjugacy is much harder to obtain, because it has to preserve the structure of all periodic orbits. See Conjugation of two circle diffeomorphisms. In particular, any homeomorphism conjugate to rotation by a rational multiple of $\pi$ must be periodic: that is, $f^n$ is the identity for some $n$. In dimensions up to $2$ the converse is true as well:

Theorem (Kerékjártó, Brouwer, Eilenberg) Every periodic homeomorphism of $S^2$ is topologically conjugate to a rigid motion of $S^2$.

This is no longer true in higher dimensions.

More results on topological classification of homeomorphisms can be found in:

1. Introduction to the Modern Theory of Dynamical Systems by Katok and Hasselblatt
2. Introduction to the Qualitative Theory of Dynamical Systems on Surfaces by Aranson, Belitskiĭ, Zhuzhoma.

Other reading:

• An introduction to rotation theory by Christian Kuehn, an accessible explanation of rotation numbers with embedded animations.
• Differentiable dynamical systems, an influential survey of the subject (not restricted to one dimension) by Stephen Smale.

Much of research in this area is motivated by qualitative theory of differential equations (dynamical systems), and therefore concerns the topological conjugacy of diffeomorphisms.