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In this paper, we consider the family of hyperbolic quadratic polynomials parametrised by a complex constant; namely Pc(z) = z 2 + c with |c| < 1 and the family of hyperbolic cubic polynomials parametrised by two complex constants; namely P (a 1 , a 0 ) (z) = z 3 +a 1 z +a 0 with |a i | < 1, restricted on their respective Julia sets. We compute the Lyapunov characteristic exponents for these polynomial maps over corresponding Julia sets, with respect to various Bernoulli measures and obtaindoi:10.3934/jcd.2019004 fatcat:upzl7xoqgjgyxk7af7y3ac5ezu