I am struggling with section 3.3 of the following thesis https://smartech.gatech.edu/xmlui/bitstream/handle/1853/29610/grigo_alexander_200908_phd.pdf. Page 21 is fine, then the problems occur in pages 22 and 23 which I believe are mainly down to notation.
"For $\epsilon_0 > 0$ consider a family of $C^5$ function $L_{\epsilon}: U \times U \rightarrow \mathbb{R^2}$ for $|\epsilon|< \epsilon_0$ which satisfy
$ \displaystyle \partial_{\epsilon}\Big|_{\epsilon=0}L_{\epsilon}(s,s_1)=C \frac{s^4+s_1^4}{24}+O_5(s,s_1)$ and $\partial_s \partial_{s_1}L_0(0,0) \neq 0$ for some $C \neq 0$."
What do the $\partial_{\epsilon}$ and $\partial_s \partial_{s_1}$ mean? What do they act on?
Then $L_{ij} := \partial^i_s \partial^j_{s_1}L_0(s,s_1)$. What are $\partial^i_s$ and $\partial^j_{s_1}$, what do $i$ and $j$ correspond to?
In particular on page 23 I cannot see how $\partial_s L_\epsilon(s, S_{\epsilon}(s,y))=y$ and $\partial_{s_1}L_{\epsilon}(s,S_{\epsilon}(s,y))=Y_{\epsilon}(s,y)$ are derived? I seem to be missing some identities used.
Finally, on page 24, I cannot see how $\partial_{\epsilon}\Big|_{\epsilon=0}A_{\epsilon}$ only the third order term $Im c_{21}$? Why is it third order, why arent the other terms third order?
@Voliar: I think you are right, at least treating $L_\varepsilon$ as a single-valued function seems consistent(I only gave a quick browse) with everything else on the mentioned pages.