Hello Tommy. I'm very impressed with your analysis and it's base of mathematical concepts. I have some feedback that hopefully will be of use. Questions 1 & 2: When I submitted a paper to the Annals of Mathematics the referee felt I should have discussed the role of germs in the study of iterated functions. Germs are sets of points with the same local properties, so they for equivalence classes. In iteration of functions the point zero is a different germ than it's immediate surrounding points.
1&2: I don't think {0,1,e,e^e,...} and their forward orbits are singularities, I think they are a different germ from their surrounding points.
3: A branch was picked when L was selected. In my research I consider the entire countable infinite set of branches indirectly when I deal with the fixed points on the different branches. For example I was able to compute the position and Lyapunov multiplier of a neighboring fixed point.
NOTE: I was wrong about {0,1,e,e^e,...} not being singularities. If they are at corners then there is a singularity in the second derivative.
1&2: I don't think {0,1,e,e^e,...} and their forward orbits are singularities, I think they are a different germ from their surrounding points.
3: A branch was picked when L was selected. In my research I consider the entire countable infinite set of branches indirectly when I deal with the fixed points on the different branches. For example I was able to compute the position and Lyapunov multiplier of a neighboring fixed point.
NOTE: I was wrong about {0,1,e,e^e,...} not being singularities. If they are at corners then there is a singularity in the second derivative.
Daniel