To be honest I have also no better idea than to call it \(\eta_-\)
I made another animation showing the fundamental region and its image.
Though its not "officially" proven I think that, as long we have a fundamental region (a region bounded by a curve \(\ell\) connecting the two fixed points and its image \(f(\ell)\)) and the function \(f\) is (holomorphic and) injective on that region then there exists the perturbed Fatou coordinates (aka Abel function) \(\Phi\) that is injective on the region and it is the only Abel function up to an additive constant. Alternatively one can also use the uniqueness criterion of Paulsen for the superfunction (inverse of the Abel function) that it tends to fixed point 1 when imaginary part goes to \(+\infty\) and to fixed point 2 if the imaginary part goes to \(-\infty\).
And I guess this not only valid for two repelling fixed points, but for any fixed point pair.
In the following animation the black line is the connection from the indifferent fixed point to the primary fixed point, call it \(\ell\). The blue line is \(b^\ell\) and the green line is \(b^{b^\ell}\). I.e. we have the fundamental region between the black and blue line and its image between the blue and green line. So we can visually check for injectivity.
So \(b^z\) seems to be injective on our chosen fundamental region up to the indifferent fixed point at \(\eta_-\), just a bit further (which is the last part of the animation) it is not anymore (if we stick to the repelling fixed point corresponding to index 1 of the LambertW function. If we change however to fixed point with index -1, then it is just the conjugated case, and everything works out fine)
I didnt find any constellation of injectivity on the fundamental region if the indices of the fixed points differ more than 1, particularly not for the two repelling fixed point close to (the indifferent fixed point) \(\eta_-\).
I made another animation showing the fundamental region and its image.
Though its not "officially" proven I think that, as long we have a fundamental region (a region bounded by a curve \(\ell\) connecting the two fixed points and its image \(f(\ell)\)) and the function \(f\) is (holomorphic and) injective on that region then there exists the perturbed Fatou coordinates (aka Abel function) \(\Phi\) that is injective on the region and it is the only Abel function up to an additive constant. Alternatively one can also use the uniqueness criterion of Paulsen for the superfunction (inverse of the Abel function) that it tends to fixed point 1 when imaginary part goes to \(+\infty\) and to fixed point 2 if the imaginary part goes to \(-\infty\).
And I guess this not only valid for two repelling fixed points, but for any fixed point pair.
In the following animation the black line is the connection from the indifferent fixed point to the primary fixed point, call it \(\ell\). The blue line is \(b^\ell\) and the green line is \(b^{b^\ell}\). I.e. we have the fundamental region between the black and blue line and its image between the blue and green line. So we can visually check for injectivity.
So \(b^z\) seems to be injective on our chosen fundamental region up to the indifferent fixed point at \(\eta_-\), just a bit further (which is the last part of the animation) it is not anymore (if we stick to the repelling fixed point corresponding to index 1 of the LambertW function. If we change however to fixed point with index -1, then it is just the conjugated case, and everything works out fine)
I didnt find any constellation of injectivity on the fundamental region if the indices of the fixed points differ more than 1, particularly not for the two repelling fixed point close to (the indifferent fixed point) \(\eta_-\).
