We define a (chaotic) deterministic variant of random multiplicative c
ascade models of turbulence. It preserves the hierarchical tree struct
ure, thanks to the addition of infinitesimal noise. The zero-noise lim
it can be handled by Perron-Frobenius theory, just like the zero-diffu
sivity limit for the fast dynamo problem. Random multiplicative models
do not possess Kolmogorov 1941 (K41) scaling because of a large-devia
tions effect. Our numerical studies indicate that deterministic multip
licative models can be chaotic and still have exact K41 scaling. A mec
hanism is suggested for avoiding large deviations, which is present in
maps with a neutrally unstable fixed point.