Talk:Saddle-node bifurcation for maps
Comments from Reviewer A:
The only criticism I have is that nothing is said about the relation between alpha and beta in the conjugacy claim (one dimensional case). Presumably you mean to say that for every alpha there exists a beta such that the two systems are conjugate? Or that there is a function beta(alpha) that is continuous (smooth?) (monotone, at least?) that matches up the systems that are conjugate.
It is not at all necessary to invoke conjugacy to settle this problem. If I were writing the article I would at least mention that there is a simple implicit function theorem argument that establishes the bifurcation, without reducing it to any normal form.
Yu.A.: Following the first remark, I have drafted a short article "conjugate maps", where a definition of conjugate parameter-dependent maps is given. There was an empty link for some time... As for the second remark, indeed IFT implies the appearance of two fixed points but the conjugacy to the normal form provides much more complete description.
I like your definition of conjugate parameter-dependent families of maps. Now your should just change the wording a little in your saddle-node article to say (in your own words) that the alpha-family of maps is conjugate to the beta-family in the sense of conjugate parameter-dependent families. This will alert a reader who knows the definition of conjugate maps, but not conjugate families, that it is worth his while to click on the link. Then I will immediately approve the paper.
While of course I agree that the conjugacy is stronger than the implicit function argument, nevertheless the implicit function argument is the main step in proving the conjugacy and the rest is rather trivial (in the one-dimensional case). My personal view is that implicit function arguments are a central organizing principle for a large chunk of mathematics and should be pointed out when they occur. But I will not insist on this.