proof of Lemma \[exp\].

Measure of the set of excluded parameters {#s3}
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The rest of the proof of the theorem goes as follows. For $n > N_0$, let $$\Delta_n = \{ a \in \Delta_{N}: \ \ \text{(R1$)_n$ and (G3$)_n$ hold}\},$$ and set $\Delta=\cap_{n\geq N}\Delta_n$. This is our parameter set in the theorem. In this section we show that $\Delta$ has positive Lebesgue measure, and $|\Delta|\to1$ as $L\to\infty$. To this end, define two parameter sets as follows: $$E_n=\{a\in\Delta_{n-1}\setminus\Delta_n\colon \text{(R$)_{n}$ fails for $f_a$}\};$$ $$E_n'=\{a\in\Delta_{n-1}\setminus\Delta_n\colon \text{(G3$)_{n}$ fails for $f_a$}\}.$$ Obviously, $\Delta_{n-1}\setminus\Delta_n\subset E_n\cup E_n'$. We show that the measures of these two sets decrease exponentially fast in $n$. Building on preliminary results in Sect.\[paradist\], in Sect.\[R1\] we estimate the measure of $E_n$. In Sect.\[R2\] we estimate the measure of $E_n'$ and complete the proof of the theorem.

\[indhyp\] [The following will be used in the argument: Let $a\in\Delta_{n-1}$. By the definition of $\Delta_{n-1}$, (R$)_{n-1}$ holds, and thus by Lemma \[derive0\], (G1$)_{n-1}$ (G2$)_{n-1}$ hold.]{}

Equivalence of derivatives and distortion in parameter space {#paradist}
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For $c \in C$ and $i\geq0$, we define $\gamma_i^{(c)}: \Delta_{N} \to S^1$ by letting $\gamma_i^{(c)}(a) = f^{i+1}_a(c)$. In this subsection we denote $c_i(a)=\gamma_i^{(c)}(a)$ and $\tau_i(a) = \frac{d
c_i(a)}{da}$.

\[trans\] Let 