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@AlexandreEremenko I google searched for "Julia set in Non holomorphic setting" but I got noting. Any way the OP question can generate wide research when we apply it to the bijection case.
@JörgNeunhäuserer [Schwartz lemma] (en.wikipedia.org/wiki/Schwarz_lemma) is a beautiful theorem with short proof but has some implications in Hyperbolic geometry: every self holomorphic map on the Disc decreass the hyperbolic meetric
@AlexandreEremenko What are some more obstacles? An obstruction I know is a possible negative Lefschetz index. BTW the analysis of fixed point is a possible framework for consideration of OP question. But what about fixed point free and orientation preserving case please see my comment to Andy Putman
@MikhailTikhomirov but $(x,y) \mapsto (x, -y)$ can not be covered by a bioholomorphism since it is orientation reversing. In fact orientation, determinant, is preserved by conjugacy. Moreover as it is mentioned by in answer and comment by prof. Eremenko and others the fixed points are not isolated, another obstruction
@DavidGao I mean: Pick a classic non compact space $Y$ then one point compactify to $X$ then perturbe $X$ to $X_q$ in the virtual word which commutativity is forbidden then try to realize $X_q$ as a minimal unitization. This is my concern.
@DavidGao Thank you for your attention to my question. Ok we can talk later if you have no time just now. Since last night I realy wonder if there would be a unital perturbation whose realization as minimal unitization of a non unital perturbation would produce some interesting stories. We possibly talk later
@DavidGao Yes you did. But my question is that why these relations generate $C_0(R^2)$? the term $\frac{1}{x^2+y^2+1}$ you idicated to remind of "Poincare compactification" which is NOT a 1 point compactification. In this case the equator of sphere play the role of infinity(Not a unique point $\infty$)
@DavidGao Any way it would be interesting to have a perturbation of a compact space which is not minimal compactification of any non compact perturbation. can one imagin a possible example?
I remember that descriptiin of classical and NC torus is inspir3d by Fourier series periodic functions in two variabkes. So how do you describe NC non compact plane?
@DavidGao thank you. what can be said about the generalization? The last part of the question? Is there a perturbation $A_q$ of a classical space X which is a compactification of a classical Y such that no perturbation of $C_0(Y)$ can be found whose unitization is $A_q$?
yes I was mistaken that was an ewxercise in rudin functional analysis. i am sorry. But by the above comment I was actually thinking to the maximality and optimality of the set of relations. That was a random relation that I pointed out to(which was wrong. Ok replace it with another one say $-log(q)ab$. BTW the optimality requirement implies that we exclude the third relation in OP question. do you agree?
