Simultaneous peripheral full-turn equations have unique small solutions

Shrink to a closed convex polydisc $B_{ϵ}=\{\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}\le ϵ\}$ inside the holomorphy domain. Its compactness and continuity of the derivatives give a common Lipschitz constant $L$ for all ${\tau }_i$ there, by integrating their derivatives along line segments. Put $A={\max }_i\mathrm{\mid }\mathrm{Im}{\tau }_i(0){\mathrm{\mid }}^{-1}$ and shrink $ϵ$ so that $ALϵ\le 1\mathrm{/}4$ and every $\mathrm{Im}{\tau }_i(u)$ stays nonzero on $B_{ϵ}$. For any nonzero real pair $(p,q)$, $p+q{\tau }_i(0)\mathrm{\ne }0$ and $\mathrm{\mid }q\mathrm{/}(p+q{\tau }_i(0))\mathrm{\mid }\le A$. Moreover $\mathrm{\mid }p+q{\tau }_i(0)\mathrm{\mid }\ge c\sqrt{p^2+q^2}$ for one common $c>0$: the real-linear maps $(p,q)\mapsto p+q{\tau }_i(0)$ are invertible, and their moduli have a positive minimum on the unit circle.

For a parameter tuple with finite pairs outside a sufficiently large radius $R$, define $a_i=q_i\mathrm{/}(p_i+q_i{\tau }_i(0))$ and the $w_i$ in the Statement. At an infinite coordinate put $a_i=w_i=0$. On $B_{ϵ}$ define $T_i(u)=w_i\mathrm{/}[1+a_i({\tau }_i(u)-{\tau }_i(0))]$. Step 1.1 makes every denominator have modulus at least $3\mathrm{/}4$, hence $\mathrm{\parallel }T(u){\mathrm{\parallel }}_{\mathrm{\infty }}\le (4\mathrm{/}3)\mathrm{\parallel }w{\mathrm{\parallel }}_{\mathrm{\infty }}$. Also $\mathrm{\parallel }T(u)-T(u^{\mathrm{\prime }}){\mathrm{\parallel }}_{\mathrm{\infty }}\le (16\mathrm{/}9)AL\mathrm{\parallel }w{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }u-u^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}$. Since step 1.1 makes $\mathrm{\parallel }w{\mathrm{\parallel }}_{\mathrm{\infty }}\le 2\pi \mathrm{/}(cR)$, choose $R$ so large that $\mathrm{\parallel }w{\mathrm{\parallel }}_{\mathrm{\infty }}\le ϵ\mathrm{/}2$ and $(16\mathrm{/}9)AL\mathrm{\parallel }w{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1\mathrm{/}2$. The closed polydisc is nonempty and complete, and $T$ maps it into itself, so [F1] supplies its unique fixed point. Its defining equations are exactly the simultaneous full-turn equations at finite coordinates, and $u_i=0$ at infinite ones. At a finite coordinate $w_i\mathrm{\ne }0$, so its solution $u_i$ cannot vanish. Every solution in $B_{ϵ}$ is such a fixed point and is therefore this one.

The fixed-point bound in step 2.1 gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}\le (4\mathrm{/}3)\mathrm{\parallel }w{\mathrm{\parallel }}_{\mathrm{\infty }}$. Its coordinate equation also gives $\mathrm{\mid }{u}_i-w_i\mathrm{\mid }\le (4\mathrm{/}3)A\mathrm{\mid }{w}_i\mathrm{\mid }\mathrm{\mid }{\tau }_i(u)-{\tau }_i(0)\mathrm{\mid }\le (16\mathrm{/}9)AL\mathrm{\mid }{w}_i\mathrm{\mid }\mathrm{\parallel }w{\mathrm{\parallel }}_{\mathrm{\infty }}$. This proves the stated uniform expansion, including every $w_i=0$ coordinate exactly.

Define the coefficient map $\mathrm{\Phi }$ on the open polydisc by the equations in the Statement. If $u_i\mathrm{\ne }0$, the real-linear equation $p_i+q_i{\tau }_i(u)=2\pi i\mathrm{/}{u}_i$ has a unique real pair, continuously depending on $u$. If $u_i\to 0$, the norms of these pairs tend to infinity uniformly with the other coordinates in $B_{ϵ}$: boundedness of ${\tau }_i$ gives $2\pi \mathrm{/}\mathrm{\mid }{u}_i\mathrm{\mid }=\mathrm{\mid }{p}_i+q_i{\tau }_i(u)\mathrm{\mid }\le C\sqrt{p_i^2+q_i^2}$. Thus $\mathrm{\Phi }$ is continuous also at its infinite coordinates, with $\mathrm{\Phi }(0)=(\mathrm{\infty },\dots ,\mathrm{\infty })$. Let $V$ be the product neighborhood specified using $R$ in step 2.1, and put $U={\mathrm{\Phi }}^{-1}(V)$ inside the open polydisc. This is an open neighborhood of $0$. Every parameter in $V$ has its unique solution in the interior by step 2.1, so $\mathrm{\Phi }:U\to V$ is bijective.

Its inverse is continuous on the whole product neighborhood, including partially unfilled tuples. If parameter tuples converge, their fixed-point maps $T$ from step 2.1 converge uniformly on $B_{ϵ}$. For a finite limiting coordinate, $a_i,w_i$ converge in the displayed rational formulas. For an infinite limiting coordinate, $w_i\to 0$ while $\mathrm{\mid }{a}_i\mathrm{\mid }\le A$, so the corresponding $T_i$ tends uniformly to zero by the $3\mathrm{/}4$ denominator bound, regardless of the directions of the real pairs. If $u,u^{\mathrm{\prime }}$ are the two fixed points and $T,T^{\mathrm{\prime }}$ their maps, then $\mathrm{\parallel }u-u^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}\le \frac{1}{2}\mathrm{\parallel }u-u^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}+{\sup }_{B_{ϵ}}\mathrm{\parallel }T-T^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}$, proving continuity. This establishes the homeomorphism. No primitivity was used in this analytic proof; primitivity enters only when a later geometric construction identifies a smooth solid-torus meridian.