Does there exist a rational function $f\in\Bbb{C}(z)$ whose Julia set coincides with $ T:=\left\{\left(x,\sin\left(\frac{1}{x}\right)\right)\,\Big|\,x\in\left(0,\frac{1}{\pi}\right]\right\}\cup(\{0\}\times[-1,1]) $? What if we require the Julia set to only be homeomorphic with $T$?

Does there exist a rational function $f\in\Bbb{C}(z)$ whose Julia set coincides with $$ T:=\left\{\left(x,\sin\left(\frac{1}{x}\right)\right)\,\Big|\,x\in\left(0,\frac{1}{\pi}\right]\right\}\cup\big(\{0\}\times[-1,1]\big)\;? $$ What if we require the Julia set to only be homeomorphic with $T$?

Does there exist a rational function $f\in\Bbb{C}(z)$ whose Julia set coincides with $ \left\{\left(x,\sin\left(\frac{1}{x}\right)\right)\,\Big|\,x\in\left(0,\frac{1}{\pi}\right]\right\}\cup(\{0\}\times[-1,1]) $?

Does there exist a rational function $f\in\Bbb{C}(z)$ whose Julia set coincides with $ T:=\left\{\left(x,\sin\left(\frac{1}{x}\right)\right)\,\Big|\,x\in\left(0,\frac{1}{\pi}\right]\right\}\cup(\{0\}\times[-1,1]) $? What if we require the Julia set to only be homeomorphic with $T$?