  We define a (chaotic) deterministic variant of random multiplicative cascade
models of turbulence. It preserves the hierarchical tree structure, thanks to
the addition of infinitesimal noise. The zero-noise limit can be handled by
Perron-Frobenius theory, just as the zero-diffusivity limit for the fast dynamo
problem. Random multiplicative models do not possess Kolmogorov 1941 (K41)
scaling because of a large-deviations effect. Our numerical studies indicate
that deterministic multiplicative models can be chaotic and still have exact
K41 scaling. A mechanism is suggested for avoiding large deviations, which is
present in maps with a neutrally unstable fixed point.
