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Von Neumann-Gale dynamical systems are defined in terms of multivaluedoperators in spaces of random vectors, possessing certain properties ofconvexity and homogeneity. A central role in the theory of such systems isplayed by a special class of paths (trajectories) called rapid: they grow overeach time period t-1,t in a sense faster than others. The paper establishesexistence and characterization theorems for such paths showing, in particular,that any trajectory maximizing a logarithmic functional over a finite timehorizon is rapid. The proof of this result is based on the methods of convexanalysis in spaces of measurable functions. The study is motivated by theapplications of the theory of von Neumann-Gale dynamical systems to themodeling of capital growth in financial markets with frictions -- transactioncosts and portfolio constraints.

Log-optimal and rapid paths in von Neumann-Gale dynamical systems