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I'm interested in understanding functions that describe themselves in a certain degree $ n $. Let me define what I mean by this:

A function $ f(x) $ is said to describe itself in degree $ n $ if there exists a function $ g_{n,f} $ with an inverse such that

$$ f = g^{-1} \circ f^n \circ g $$

Notably, every function describes itself in degree 1, since $ g(t) = t $ in that case. An example to illustrate this concept is the function $ f(x) = 2x $, which describes itself in degree 2 with $ g(x) = a|x|x $ and its inverse $ g^{-1}(x) = \frac{\sqrt{|\frac{x}{a}|} \cdot |\frac{x}{a}|}{(\frac{x}{a})} $ for $ x \neq 0 $ and 0 when $ x = 0 $, where $ a $ is a non-zero real number.

Through this exploration, I've found that $ g(x) $ for n degree is related to $ a \cdot x^n $, but I'm struggling to find a $ g(x) $ for $f(x) = x^2$ for any other degree other than the trivial case of degree equaling $ 1 $.

my main question is this What types of functions describe themselves with degree $ n $?

My follow up questions that I think are important as well are as follows:

  1. How does this change if we restrict our functions to complex-to-complex, real-to-real, positive-to-positive, or integer-to-integer mappings?
  2. What are the constraints required for a function to describe itself in a specific degree $ n $?
  3. Are there functions that describe themselves in some $ n > 1 $ but not in all such $ n $?
  4. Does the function $ x^2 $ describe itself in any degree other than degree 1?

I appreciate any insights or references that could help in understanding these types of functions and their properties.

11123469131928Plastic
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  • The notation isn't easy to process. You are basically asking for solutions of $f=g^{-1}!\circ ! f^n \circ ! g$ with $n \in \mathbb{N}$, right? – David M Dec 31 '23 at 05:47
  • @DavidM yes you are right that is what I mean – 11123469131928Plastic Dec 31 '23 at 05:51
  • Your question will probably get more attention if you edit it along those lines to make it easier to read. – David M Dec 31 '23 at 05:54
  • @DavidM do you have any other suggestions how I can make it more readable? – 11123469131928Plastic Dec 31 '23 at 05:58
  • Perhaps make the rest of the question compatible with the notation introduced in the edit you have just made. In other words, get rid of things like $g_{2,2x}$ and whatnot. – David M Dec 31 '23 at 06:01
  • Is this relevant? Idempotence – Kartik Pandey Dec 31 '23 at 06:04
  • @kartikPandey honestly I don’t think so, if f(x)= x it would trivially describe itself with all n degrees, though if f(f(x)) = x then it would be more interesting, when you apply the function twice you get the same number back so it oscillates. for degree n = 2m g(x) = x, for degree n = 2m+1 g(x) = f(x). but I don’t see how Idempotence is relevant but I might be wrong. – 11123469131928Plastic Dec 31 '23 at 06:51

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