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Q1. Take $X=\{\\,z\in\mathbb C\mid 1<|z|<2\\,\}$ and $f(z)=i \frac{z}{\sqrt{|z|}}$$f(z)=-\tfrac{z}{\sqrt{|z|}}$.
It seems to answer the other questions.
You may take a spiral in X which is $f$-invariant. This produce a simply connected example.
Q1. Take $X=\{\\,z\in\mathbb C\mid 1<|z|<2\\,\}$ and $f(z)=i \frac{z}{\sqrt{|z|}}$.
Q1. Take $X=\{\\,z\in\mathbb C\mid 1<|z|<2\\,\}$ and $f(z)=-\tfrac{z}{\sqrt{|z|}}$.
Q1. Take $X=\{\\,z\in\mathbb C\mid 1<|z|<2\\,\}$ and $f(z)=i\cdot z/\sqrt{z}$$f(z)=i \frac{z}{\sqrt{|z|}}$.
Q1. Take $X=\{\\,z\in\mathbb C\mid 1<|z|<2\\,\}$ and $f(z)=i\cdot z/\sqrt{z}$.
