CRITICALLY FINITE RANDOM MAPS OF AN INTERVAL

Given a finite collection G of closed subintervals of the unit interval [0, 1] with mutually empty interiors, we consider random multimodal C-3 maps with negative Schwarzian derivative, mapping each interval of G onto the unit interval [0,1]. The randomness is governed by an invertible ergodic map theta : Omega -> Omega preserving a probability measure m on some probability space Omega. We denote the corresponding skew product map by T and call it a critically finite random map of an interval. We prove that there exists a subset AA(T), defined in Definition 9.1, of [0, 1] with the following properties: 1. For each t is an element of AA(T) a t-conformal random measure nu(t) exists. We denote by lambda(t,nu t,omega) the corresponding generalized eigenvalues of the corresponding dual operators L-t,(omega)*, omega is an element of Omega. 2. Given t >= 0 any two t-conformal random measures are equivalent. 3. The expected topological pressure of the parameter t: epsilon P(t) := integral(Omega) log lambda(t,nu,omega)dm(omega). is independent of the choice of a t-conformal random measure nu. 4. The function AA(T) (sic) t bar right arrow epsilon P (t) is an element of R is monotone decreasing and Lipschitz continuous. 5. With b(T) being defined as the supremum of such parameters t is an element of AA(T) that epsilon P(t) >= 0, it holds that epsilon P(b(T)) = 0 and [0, b(T)] subset of Int(AA(T)). 6. HD(J(omega)(T)) = b(T) for m-a.e omega is an element of Omega, where J(omega)(T), omega is an element of Omega, form the random closed set generated by the skew product map T. 7. b(T) = 1 if and only if boolean OR(Delta is an element of g) Delta = [0, 1], and then J(omega)(T) = [0,1] for all omega is an element of Omega.