The Global Statistics of Return Times: Return Time Dimensions Versus Generalized Measure Dimensions

We investigate return times in dynamical systems, i.e. the time required by a trajectory to complete a return journey to a neighborhood of the initial position. In particular, we study the relations holding between the scaling exponents of phase-space moments of return times in balls of diminishing radius, on the one side, and the generalized dimensions of invariant measures, on the other. Because of a heuristic use of Kac theorem, the former have been used in place of the latter in numerical and experimental investigations: to mark the distinction, we call them return time dimensions. We derive a full set of inequalities linking generalized dimensions of invariant measures and return time dimensions. We comment on their optimality with the aid of two maps due to von Neumann–Kakutani and to Gaspard–Wang. We conjecture a formula for the return time dimensions in a typical system. We only assume that the dynamical system under investigation is ergodic and that motion takes place in a compact, finite dimensional space.