Clustering of exponentially separating trajectories

It might be expected that trajectories for a dynamical system which has no negative Lyapunov exponent (implying exponential growth of small separations will not cluster together. However, clustering can occur such that the density $ρ(Δx)$ of trajectories within distance $Δx$ of a reference trajectory has a power-law divergence, so that $ρ(Δx)\sim Δx^{-β}$ when $Δx$ is sufficiently small, for some $0<β<1$. We demonstrate this effect using a random map in one dimension. We find no evidence for this effect in the chaotic logistic map, and argue that the effect is harder to observe in deterministic maps.

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