Julia sets of the random iteration systems and its subsystems

Suppose thatP (p ₁,p ₁, …,p _M ) is a probability vector withp _i > 0 and Y = {1, 2, …, M}. Let (Y, 2^Y, μ) be a probability space withμ(i) =p _i ,i = 1, 2, …,M, and (Σ_M, ℬ,m) =Π ₀ ^∞ (Yt 2 ^U ,μ). It is shown that for any a (0≤a ≤ 1), there exists a setU ∈ B such thatm (U) = a and the Julia set associated withU is equal to the Julia set associated withΣ _M . Moreover repelling fixcd polnts with respect toU are dense in the Julia set associated withU.