fractional iterates of arg(z) and |z|

[JmsNxn]

Is it possible to define a function
h(h(z)) = |z|, h(z) =/= |z|

or
h(h(z)) = arg(z)

I wonder because these seem like difficult functions to crack, considering they have no Taylor series.

[BenStandeven]

Is it possible to define a function
h(h(z)) = |z|, h(z) =/= |z|

or
h(h(z)) = arg(z)

I wonder because these seem like difficult functions to crack, considering they have no Taylor series.

We could set h(z) = |z| for z with non-negative real part, h(z) = -iz for z with negative real and non-negative imaginary part, and h(z) = iz for z with negative and imaginary parts. Then we'd get h(h(z)) = |z| for all z. A similar technique should work for h(h(z)) = arg(z).

[JmsNxn]

that was easy