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This accepted answer to the question about $BU_\otimes$ made me recall that I want to read about this general phenomenon for a long time.

What will follow is sort of vernacular but whether it can be translated into decent language is actually part of my question :D

So the double quoted words are all wrong but believed by me to be replaceable by something legitimate, and among other things I am asking whether this is the case and what the correct words must be.

So let us intuitively look at ring spectra as "actual" "spaces" with addition, multiplication and unit element satisfying ring identities "up to higher homotopies".

Such a guy $R$ has $R_0$ - the "connected component of 0" and $R_1$ - the "connected component of 1".

I know two examples when there is an "exponential map" $R_0\to R_1$ which is moreover an "isomorphism" between the additive structure of $R_0$ and multiplicative structure of $R_1$. Their "infinite loop space representatives" are $\Omega^\infty S^\infty$ and $\mathbb Z\times BU$ respectively, so although existence of these maps in these cases is seemingly very deep and important fact, in a sense it is trivial - by definition all connected components are just the same (well not actually trivial in the first case - it only becomes as trivial as the second after applying the highly nontrivial Barratt-Priddy equivalence $\Omega^\infty S^\infty\simeq\mathbb Z\times B\Sigma_\infty^+$).

Now the main part of my question.

Where can I read about a systematic study of such exponential maps (not necessarily isomorphisms) for various ring spectra?

Later - as Charles Rezk says in the comment below, it does not make any sense to call this map exponential since it does not transform addition into multiplication at all.

I am still not sure whether I should withdraw the question altogether or in fact I still had to ask something sensible after all. Let me think some more...

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    $\begingroup$ I don't understand your examples. In neither case is there an equivalence between $R_0$ (with additive structure) and $R_1$ (with multiplicative structure) which is compatible with the given $H$-space structures. $\endgroup$
    – Charles Rezk
    Commented Jun 3, 2014 at 14:18
  • $\begingroup$ It seems that there is a kind of logarithm, though: ncatlab.org/nlab/show/logarithmic+cohomology+operation $\endgroup$
    – Qiaochu Yuan
    Commented Jun 3, 2014 at 19:37
  • $\begingroup$ Mamuka, do you mean something like this toy commutative algebra example? Let $(R,\frak m)$ be a local ring with $\frak m^2=0$. Then there is an injective homomorphism ${\frak m}\rightarrow R^\times\colon x\mapsto 1+x$. In this case, you would like to study the `connected component of $1$ in the spectrum of unit of $R$'. For such a thing, I believe you have different versions in the literature, and it's a matter of current research to dilucidate which is the most appropriate one (I may be saying something silly here, since I'm not an expert, but this is what I've understood from experts). $\endgroup$
    – Fernando Muro
    Commented Jun 5, 2014 at 14:36
  • $\begingroup$ @Fernando Yes yes thanks, something like this, and in this case $x+y$ does indeed go to $(1+x)(1+y)$, but this is quite rare of course... $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jun 5, 2014 at 14:56
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    $\begingroup$ For $KO$ (completed at $p$), the Rezk logarithm is a map $gl_1 KO \to KO$. By Theorem 4.11 of math.uiuc.edu/~mando/papers/koandtmf.pdf, this map is a weak equivalence on 3-connected covers. This means that you get an "exponential" map $KSO(X) \to (1+KSO(X))^\times$, which seems like something you might be interested in. (This is probably in Rezk's logarithm paper.) $\endgroup$
    – skd
    Commented Jul 12, 2017 at 21:09

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