Absolutely continuous invariant measures for random maps with position dependent probabilities (Q1869958)

Let \((X,{\mathcal B},v)\) be a probability space, \(X= [0,1]\). Let \(\tau_i: [0,1]\to [0,1]\), \(i= 1,\dots, K\), be point transformations and define the random map \(\tau\) by choosing \(\tau_k\) with probability \(p_k\), \(p_k> 0\), \(\sum_k p_k= 1\). A measure \(\mu\) on \([0,1]\) is called invariant under \(\tau\) if \(\mu(A)= \sum_k p_k\mu(\tau^{-1}_k A)\), \(A\in{\mathcal B}\). Define the transition function for the random map \(T= \{\tau_1,\dots, \tau_K; p_1(x),\dots, p_K(x)\}\) as follows: \(P(x, A)= \sum_k p_k(x)\chi_A(\tau_k(x))\), \(A\in{\mathcal B}\). The transition function \(P\) induces an operator \(P_*\) on measures on \((X,{\mathcal B})\) defined by \(P_*\mu= \int P(x,A) d\mu(x)\). If \(\mu\) has a density \(f\) with respect to \(v\), then \(P_*\mu\) has a density denoting by \({\mathbf P}_Tf\). The measure \(\mu\) is invariant under \(T\) iff \(f\) is invariant under \({\mathbf P}_T\). The main results of the paper: 1. Under some assumption the random map \(\tau\) has an absolutely continuous measure \(\mu\). 2. A method of approximating the fixed point of \({\mathbf P}_T\) by fixed points of the matrix operator is described. 3. If \(f_k\) is the density of \(\tau_k\), \(a_k> 0\), \(k= 1,\dots, K\), the density \(f= a_1f_1+\cdots+ a_kf_k\) is invariant under \(T= \{\tau_1,\dots, \tau_K; p_1,\dots, p_K\}\) with \(p_k= {a_k f_k\over a_1f_1+\cdots+ a_kf_k}\). 4. If \(\Gamma= \{\tau_1,\dots, \tau_k\}\), \({\mathcal A}_\Gamma\) is the set of attainable densities, then \({\mathcal A}_\Gamma\) is closed in the weak and norm topologies in \(L^1\). In the paper there are examples, numerical experiments and open questions.