asymmetric damage segregation (ads) is ubiquitous among unicellular organisms: after a mother cell divides, its two daughter cells receive sometimes slightly, sometimes strongly different fractions of damaged proteins accumulated in the mother cell. previous studies demonstrated that ads provides a selective advantage over symmetrically dividing cells by rejuvenating and perpetuating the population as a whole. in this work we focus on the statistical properties of damage in individual lineages and the overall damage distributions in growing populations for a variety of ads models with different rules governing damage accumulation, segregation, and the lifetime dependence on damage. we show that for a large class of deterministic ads rules the trajectories of damage along the lineages are chaotic, and the distributions of damage in cells born at a given time asymptotically becomes fractal. by exploiting the analogy of linear ads models with the iterated function systems known in chaos theory, we derive the frobenius–perron equation for the stationary damage density distribution and analytically compute the damage distribution moments and fractal dimensions. we also investigate nonlinear and stochastic variants of ads models and show the robustness of the salient features of the damage distributions.