Attainable densities for random maps

Abstract

We consider a random map T = T ( Γ , ω ) , where Γ = ( τ 1 , τ 2 , … , τ K ) is a collection of maps of an interval and ω = ( p 1 , p 2 , … , p K ) is a collection of the corresponding position dependent probabilities, that is, p k ( x ) ⩾ 0 for k = 1 , 2 , … , K and ∑ k = 1 K p k ( x ) = 1 . At each step, the random map T moves the point x to τ k ( x ) with probability p k ( x ) . For a fixed collection of maps Γ, T can have many different invariant probability density functions, depending on the choice of the (weighting) probabilities ω. Most of the results in this paper concern random maps where Γ is a family of piecewise linear semi-Markov maps. We investigate properties of the set of invariant probability density functions of T that are attainable by allowing the probabilities in ω to vary in a certain class of functions. We prove that the set of all attainable densities can be determined algorithmically. We also study the duality between random maps generated by transformations and random maps constructed from a collection of their inverse branches. Such representation may be of greater interest in view of new methods of computing entropy [W. Słomczyński, J. Kwapień, K. Życzkowski, Entropy computing via integration over fractal measures, Chaos 10 (2000) 180–188].