Non-ergodic maps in the tangent family

AbstractWe consider maps in the tangent family for which the asymptotic values are eventually mapped onto poles. For such functions the Julia set J(f) = C. We prove that for almost all z ∈ J(f) the limit set w(z) is the post-singular set and f is non-ergodic on J(f). We also prove that for such f does not exist a f-invariant measure absolutely continuous with respect to the Lebesgue measure finite on compact subsets of C