Properties of distributions and correlation integrals for generalised versions of the logistic map

We study a generalised version of the logistic map of the unit interval $(0,1)$, in which the point $x$ is taken to $1-|2x-1|^nu$. Here, $nu >0$ is a parameter of the map, which has received attention only when $nu =1$ and 2. We obtain the invariant density when $nu = frac12$, and derive properties of invariant distributions in all other cases. These are obtained by a mixture of analytic and numerical argument. In particular, we develop a technique for combining "parametric" information, available from the functional form of the map, with "non-parametric" information, from a Monte Carlo study. Properties of the correlation integral under the invariant distribution are also derived. It is shown that classical behaviour of this test statistic, which demands that the logarithm of the integral have slope equal to the lag, is valid if and only if $nu leq 2$.