The von Mises distribution is entirely defined on the circle with a density given by $$f(x) = (2\,\pi\, I_0(\kappa))^{-1} \exp(\kappa \cos(x-\mu))\ ,$$ where $x$ is in an arbitrary real interval of length $2\pi$, $I_0$ is the Bessel-I function of order 0, $\mu$ is a location parameter, and $\kappa>0$ is a concentration parameter.
My question is: What transformations of a von Mises random variable preserve the distribution family?
A slightly simplified version of this question (considering an important special case) can be restated as follows: Is there a bijective function (defined on the circle) $T$ and a $\kappa>0$ solving $$(I_0(1))^{-1}\, \left|T'(x)\right|\, \exp(\cos(T(x))) = (I_0(\kappa))^{-1} \exp(\kappa \cdot \cos(x))\ ,$$
The answer is easy for $\kappa=1$, because $T$ might be chosen as the identity, a simple shift, or a suitable reflection, Thus, I am interested if this can be solved when $\kappa\neq 1$. Unfortunately, I am not aware whether there are any results available on this issue.
