Properties of distributions and correlation integrals for generalized versions of the logistic map (Q1805796)

Consider the parametric function \(f(x)=f_\nu (x)=1-|2x-1|^\nu \) on \((0,1)\), with \(\nu >0\). The properties of invariant distributions are well known for \(\nu =1\) and \(2\). Recall that the distribution of a random variable \(X\) is (\(f\)-)invariant if both \(X\) and \(f(X)\) have the same distribution. The authors examine the other values of \(\nu \). They show that a non-degenerate invariant measure does not exist for small values of \(\nu \), but exists for \(\nu >\nu_0\) where \(\nu_0 <\tfrac 12\) and is close to \(\tfrac 12\). For \(\nu =\tfrac 12\) the only invariant distribution whose distribution function on \((0,1)\) is analytic, is that with density \(g(x)=2 - 2x\). Only for \(\nu \geq \tfrac 12\) the invariant distribution can be supported by the whole interval. An invariant density \(g=g_\nu \) is bounded for \(\tfrac 12<\nu <1\), but is unbounded for \(\nu >1\). In the latter case \(g(x)\sim c_1x^{(1/\nu)-1}\) and \(g(1-x)\sim c_2x^{(1/\nu)-1}\), as \(x\) approaches 0; here \(c_1,c_2>0\) are constants depending on \(\nu \). This section contains also some numerical results. The second part of the paper contains a result describing the behaviour of the correlation integral in the case of a stochastic sequence \(\{X_i\}\).