We construct a (chaotic) deterministic variant of random multiplicative cascade models of turbulence. It preserves the hierarchical tree structure, thanks to the addition of infinitesimal noise or finite-state Markov approximations of chaotic maps. The zero-noise limit can be handled by Perron-Frobenius theory, just as the zero-diffusivivity limit for the fast dynamo problem. We prove also the absence of phase transitions in conservative random multiplicative cascade models, corresponding to the non divergence of statistical moments.