ds.dynamical systems - How to construct a topological conjugacy?

Let me throw in some speculations based on my limited involvement in dynamical systems.

The conjugation formula $f=h^{-1}gh$ is in general not a type of a functional equation that can be solved by iterative approximations or a clever fixed-point trick. The problem is that you cannot determine how badly a particular $h$ fails by looking at the difference between LHS and RHS of this equation. The obstructions are not local and you don't see them until you consider all iterations of $f$ and $g$. Sometimes you can do approximations (e.g. Anosov system more or less survive under perturbations) but this works only in special types of systems (some kind of "hyperbolicity" is needed).

It seems that the only "general" way for constructing a topological conjugation is to study the orbits of $f$ and $g$ and send each orbit of $f$ to a similar orbit of $g$ so that the whole map is continuous. (A dense set of orbits is sufficient, e.g. the set of periodic points of an Anosov system.) The problem is, of course, that the structure of orbits can be really complicated. But if it is simple, one can hope to construct a conjugation directly.

For example, consider two homeomorphisms $f,g:mathbb Rtomathbb R$ satisfying $f(x)>x$ and $g(x)>x$ for all $x$. They are conjugate. To see this, consider an orbit $dots,x_{-1},x_0,x_1,x_2,dots$ of a point $x_0$ under the iterations of $f$. This is an increasing sequence and the intervals $[x_i,x_{i+1}]$ cover $mathbb R$. Every other orbit "interleaves" with this one: for example, if $y_0in(x_0,x_1)$, then $y_i:=f^i(y_0)$ lies between $x_i$ and $x_{i+1}$. So every orbit has a unique member in the interval $[x_0,x_1)$. In a sense, this interval (or rather the closed one with the endpoints glued together) naturally represents the set of all orbits.

So take any orbit $(x_i')$ of $g$ and let $h_0$ be any order-preserving bijection from $[x_0,x_1]$ to $[x_0',x_1']$. This defines a unique conjugacy map $h:mathbb Rtomathbb R$ such that $h|_{[x_0,x_1]}=h_0$: the orbit ${f^i(y)}$ of a point $yin [x_0,x_1]$ is mapped to the $g$-orbit ${g^i(h_0(y))}$ of the point $h_0(y)$. And all conjugations can be obtained this way.

Already in this simple example, you can see how fragile things can be. Even if $f$ and $g$ are smooth and have bounded derivatives, you have no control over how big the derivatives of $h$ can grow. (And you actually lose smoothness if you do the same on a closed interval rather than $mathbb R$.)

If you vary the map $g$, it remains conjugate to $f$ while the condition $g(x)>x$ holds true. But it suddenly stops being conjugate once a fixed point appears. However trivial this fact is, is shows that limit of conjugacy maps does not make sense in general.

The exercise you mention can be solved in a similar fashion as my toy example; the orbits are not much more complicated. In fact, given any two homeomorphisms $mathbb Rtomathbb R$, it is easy to understand whether they are conjugate or not (just study the intervals between fixed points). But the next step - homeomorphisms of the circle - is much more difficult: there are beautiful theorems, unexpected conterexamples, connections to number theory and other signs of a rich theory around such a seemingly trivial object. See Denjoy theorem and especially its smoothness requirements to get an idea how interesting these things can be.