What transformations preserve the von Mises distribution? 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.
This is a crosspost from Mathoverflow.
 A: Obviously $\mu$ is a location parameter, meaning that translations of the variable preserve the family.
Focus now on the shape parameter $\kappa$.  Consider any family $\Omega=\{F_\theta|\theta\in\Theta\}$ of continuous distributions.  By virtue of this continuity, whenever $X\sim F_\theta$ and $0\le q\le 1$,
$$\Pr(F_\theta(X)\le q) = q.$$
The transformation
$$G_{\theta^\prime,\theta}(X) = F_{\theta^\prime}^{-1}(F_\theta(X))$$
maps any such random variable into $Y = G_{\theta^\prime,\theta}(X)$ and
$$\Pr(Y \le y) = \Pr(F_{\theta^\prime}^{-1}(F_\theta(X)) \le y) = \Pr(F_\theta(X) \le F_{\theta^\prime}(y))=F_{\theta^\prime}(y)$$
shows that $Y \sim F_{\theta^\prime}.$
The question, then, is whether the family $\{G_{\theta^\prime,\theta}| \theta\in\Theta, \theta^\prime\in\Theta\}$ is closed under composition.  Suspecting that it should not be for the shape family of the Von Mises distribution (with $\mu=0$ and $\theta=\kappa$), I numerically searched for a solution $(\alpha,\beta)$ to the equation
$$ G_{\alpha,\beta} = G_{2,1} \circ G_{1/2,1}$$
by minimizing the $L^2$ norm between the two sides.  The difference between the best solution (with $\alpha=2.96234\ldots$ and $\beta = 2.48773\ldots$) and the right hand side is small but so clear that I doubt there was an error in the calculation.

Consequently the answer to the question--as understood in the sense described here--appears to be that only the translations (modulo $2\pi$) and, of course, the reflections $x\to a-x \mod 2\pi$ preserve the entire family of Von Mises distributions.
