Computer generated images of dynamical sets (such as Julia sets) play a prominent role in establishing new results. Roughly speaking, computational complexity of such set measures how hard it is to draw an approximation of the set with a given accuracy using a computer. It is known that already the class of quadratic polynomials contains examples with non-computable Julia sets (for them there is no algorithm which could draw an approximation with arbitrary precision) and examples with Julia sets of arbitrarily high computational complexity. In this talk I will show that for a large class of rational maps (satisfying Exponential Shrinking Condition) including almost all real polynomials $p_c (z) = z^2+c$; $c \in \mathbb R$, the Julia set has polynomial time complexity. The talk is based on a joint work with Michael Yampolsky.