Background: One says that continuous maps $f: X \to X, g: Y \to Y$ are topologically conjugate if there exists a homeomorphism $h: X \to Y$ such that $h \circ f = g \circ h$. There are many ways one can see that two maps are not topologically conjugate. For instance, if $f$ has fixed points and $g$ does not (more generally if the same power $f$ and $g$ have different numbers of fixed points), they cannot be topologically conjugate. Topological entropy provides a fancier invariant in terms of coverings (assuming $X$ and $Y$ are compact spaces).

I see also that there are many general theorems of that allow one to conclude that two maps are topologically conjuage (e.g. the Hartman-Grobman theorem).

However, I am curious:

Given two discrete dynamical systems, is there a trick one can use to construct a topological conjugacy between them?

I suppose an analogy would be a comparison between the Brouwer and Banach fixed point theorems. I'm curious if there is an iterative process as in the proof of the latter.

(Full disclosure: I was motivated to ask this question because I got stuck on what should be a simple exercise in a book on dynamical systems. The exercise was to prove that any two $C^1$ maps $f,g: \mathbb{R} \to \mathbb{R}$ whose derivatives at zero are between $0$ and $1$ are locally topologically conjugate.)

Background: One says that continuous maps $f: X \to X, g: Y \to Y$ are topologically conjugate if there exists a homeomorphism $h: X \to Y$ such that $h \circ f = g \circ h$. There are many ways one can see that two maps are not topologically conjugate. For instance, if $f$ has fixed points and $g$ does not (more generally if the same power $f$ and $g$ have different numbers of fixed points), they cannot be topologically conjugate. Topological entropy provides a fancier invariant in terms of coverings (assuming $X$ and $Y$ are compact spaces).

I see also that there are many general theorems of that allow one to conclude that two maps are topologically conjugate (e.g. the Hartman-Grobman theorem).

However, I am curious:

Given two discrete dynamical systems, is there a trick one can use to construct a topological conjugacy between them?

I suppose an analogy would be a comparison between the Brouwer and Banach fixed point theorems. I'm curious if there is an iterative process as in the proof of the latter.

(Full disclosure: I was motivated to ask this question because I got stuck on what should be a simple exercise in a book on dynamical systems. The exercise was to prove that any two $C^1$ maps $f,g: \mathbb{R} \to \mathbb{R}$ such that $0<f'(0)<1, 0 < g'(0)<1$ are locally topologically conjugate.)