Probability Density Functions of Some Skew Tent Maps

We consider a family of chaotic skew tent maps. The skew tent map is a two-parameter, piecewise-linear, weakly-unimodal, map of the interval Fa;b. We show that Fa;b is Markov for a dense set of parameters in the chaotic region, and we exactly nd the probability density function (pdf), for any of these maps. It is well known, [1], that when a sequence of transformations has a uniform limit F , and the corresponding sequence of invariant pdf's has a weak limit, then that invariant pdf must be F invariant. However, we show in the case of a family of skew tent maps that not only does a suitable sequence of convergent sequence exist, but they can be constructed entirely within the family of skew tent maps. Furthermore, such a sequence can be found amongst the set of Markov transformations, for which pdf's are easily and exactly calculated. We then apply these results to exactly integrate Lyapunov exponents.