What causes cultural stability? Culture can be studied as an evolving system, and the comparison between biological and cultural evolution has inspired a productive research agenda in which cultural stability is commonly attributed to the existence of mechanisms of high-fidelity cultural transmission. Other researchers have argued that no such copying processes are necessary to explain cultural stability, and that stability can also emerge as a by-product of convergent transformation (in which an item causes the production of another item whose form tends to deviate from that of the original item in a non random way). To investigate this issue, we present a series of stochastic simulation models of cultural evolution that make no prior assumptions about the type of processes by which cultural units propagate through a population. Results show that cultural stability can emerge and be maintained by convergent transformation alone, even in the absence of any form of copying or selection process. We also show that high-fidelity copying and convergent transformation are, contrary to some previous arguments, not opposing forces, and can in fact jointly contribute to cultural stability. Finally, we analyse how convergent transformation and high-fidelity copying can have different evolutionary signatures at the level of the population, and hence how their differing effects can be distinguished in the empirical record. Our models can be read as formalisations of Cultural Attraction Theory.