# What is the name of this relationship between two functions?

Two functions $$f$$ and $$g$$ are called ____ (the word I'm asking for) if there exists a function $$h$$ such that $$f \circ h = h \circ g$$.

The example I have in mind is the functions $$f(x) = 2x^2-1$$, $$g(x) = 2x$$ and $$h(x) = \cos(x)$$.

Another example is $$f(x) = x^2$$, $$g(x) = 2x$$ and $$h(x) = e^x$$.

I thought they were called "adjoint functions", but googling doesn't give enough information on that.

Does anyone know what the exact name is? Any reference on this topic will be equally welcome.

• There is also a possibility that there is no established name for that at all, for the reason that the property may not be important. :)
– Yes
Nov 19, 2015 at 14:44
• Oh... possibly. I thought it was something useful, for example in analysing dynamic systems. Nov 19, 2015 at 14:47
• But take not my comment the wrong way; I just thought about that and am glad to know it if it exists.
– Yes
Nov 19, 2015 at 14:49
• Yes, I got your point. Thanks. Let's see if there's an answer. Nov 19, 2015 at 14:51
• If $h$ was invertible, I would call $f$ and $g$ similar. Nov 19, 2015 at 16:55

in KCG Chapter 8 is called On Conjugacy, that being your relation. It is usually too much to expect anything global; in section 8.5 he begins to consider this relation in formal power series in $\mathbb C.$ More important than you would think. If the three series involved all have positive radius of convergence, he emphasizes this by saying analytically conjugate.
An example, from Milnor, Dynamics in One Complex Variable: Let $0$ be a fixpoint of $f,$ holomorphic in a neighborhood. Take $$f(z) = \lambda z + a_2 z^2 + a_3 z^3 + \cdots$$ Koenigs Linearization: if $|\lambda| \neq 0,1$ there is a local holomorphic change of coordinate $\phi$ with $\phi(0)=0,$ such that $$\phi f \phi^{-1}(w) = \lambda w$$
If $h$ is invertible, then $f$ and $g$ lie in the same orbit under the action of 'conjugation by $h$'. This is often shorted to $f$ is conjugate to $g$. There may be a notion of or orbit or conjugate that generalizes well when you're not in a full group, but I'm no algebraist.