Attainable densities for random maps (Q820014)

A random map \(T=T(\Gamma, \omega)\) is given by a collection of interval maps \(\Gamma =(\tau_1,\dots \tau_k)\) and a collection of position-dependent probabilities \(\omega = (p_1,\dots , p_k)\), i.e., \(p_i (x)\geq 0\), \(\sum_{i=1}^k p_i (x) =1\). For any point \(x\) of the interval \(T(x)\) is equal to \(\tau_i (x) \) with probability \(p_i (x)\). Given a collection \(\Gamma =(\tau_1,\dots ,\tau_k)\) of maps (in the paper, piecewise linear semi-Markov maps are considered), the set of corresponding attainable densities is defined as the set of functions \(f\) such that there exist probabilities \(\omega = (p_1,\dots , p_k)\), so that \(f\) is an invariant density of \(T(\Gamma, \omega)\). The authors investigate properties of this set when probabilities belong to a certain class of functions. The results enable us to decide algorithmically whether a given density is attainable for certain random maps.