In the realization of a dynamical system on a computer, all computational processes are of a discretization, where continuum state space is replaced by the finite set of machine arithmetic. When chaos is present, the discretized system often manifests collapsing effects to a fixed point or to short cycles. These phenomena exhibit a statistical structure which can be modelled by random mappings with an absorbing centre. This model gives results which are very much in line with computational experiments and there appears to be a type of universality summarized by an Arcsine Law. The effects are discussed with special reference to the family of mappings $f_e(x) = 1 - |1 - 2x|^l, x \in [0, 1], 1 \lt l \leq 2$. Computer experiments display close agreement with the predictions of the model.