In Sect.\[s2.4\] we study global dynamical properties of maps satisfying these conditions.

Elementary facts {#s2.1}
----------------

For $\varepsilon > 0$, we use $C_{\varepsilon}$ and $S_\varepsilon$ to denote the $\varepsilon$-neighborhoods of $C$ and $S$ respectively. The distances from $x \in S^1$ to $C$ and $S$ are denoted as $d_C(x)$ and $d_S(x)$ respectively. We take $L$ as a base of ${\rm log}(\cdot)$.

\[derivative\] There exist $K_0>1$ and $\varepsilon_0 >0$ such that the following holds for all sufficiently large $L$ and $f=f_{a}$:

-   for all $x\in S^1$, $$K_0^{-1}L
    \frac{d_C(x)}{d_S(x)} \leq|f'x| \leq K_0L
    \frac{d_C(x)}{d_S(x)}, \ \ \ \ \ \  |f''x| \leq
    \frac{K_0L}{d^2_S(x)};$$

-   for all $\varepsilon
    >0$ and $x \not \in C_{\varepsilon}$, $ |f'x|\geq K_0^{-1} L\varepsilon$;

-   for all $x \in C_{\varepsilon_0}$, $K^{-1}_0 L < |f''
    x| < K_0 L$.

This lemma follows immediately from $$f'=1+L \cdot \frac{\Phi'}{\Phi};
 \ \ \ \ \
f''=L \cdot \frac{\Phi''\Phi-
(\Phi')^2}{\Phi^2}$$ and our assumptions on $\Phi$ in the beginning of the introduction.

Bounded distortion {#distortion}
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Let $c\in C$, $c_0=fc$, and $n\geq1$. Let $$\label{Theta}
D_n(c_0)=\frac{1}{\sqrt{L}}\cdot\left[\sum_{i=0}^{n-1}
d_i^{-1}(c_0)\right]^{-1} \ \ \ \ \text{where} \ \ \ \ \
d_i(c_0)=\frac{d_C(c_i)\cdot
d_S(c_i)}{J^i(c_0)}.$$

\[dist\] For all $x,y \in[c_0-D_n(c_0),c_0+D_n(c_0)]$ we have $J^n(x)\leq 2J^n(y),$ provided that $c_i\notin C\cup S$ for every $0\leq i<n$.

Write 