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Is there an established name for the relationship between functions $f(x)$ and $g(x)$ satisfying $T(f(x))=g(T(x))$ for some $T(x)$?

It is similar to an isomorphism and it seems that it might be useful since it follows by induction that $T(f^n(x))=g^n(T(x))$.

One thing that this formula can be used for is a kind of fun derivation of the formula for the sum of a geometric series. If f(x)=ax+1 then $1+a+a^2+...+a^n = f^n(1)$. Let g(x)=ax. Then if T(x)= x +1/(a-1) we get $T(f(x))=g(T(x))$ and $f^n(1)+1/(a-1)=a^n(1+1/(a-1))$. Solving for $f^n(1)$ gives the formula.

user1153980
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    If $T$ is a surjection, then I believe $f$ and $g$ are called semiconjugate. This is a property that is seen frequently in dynamics, in particular, topological and complex dynamics. – Maths Matador Feb 23 '23 at 16:36
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    I've seen the term "intertwining operator" used to describe operators with the same property in more specific contexts. Intertwining function may be a good rough and ready term for it. – Charles Hudgins Feb 24 '23 at 04:44

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