Kaplan–Yorke conjecture for information dimension

Determine whether the Kaplan–Yorke dimension equals the information dimension of the invariant measure for general dissipative chaotic dynamical systems, beyond the special cases in which this equality has been proved.

Background

The paper introduces the Kaplan–Yorke dimension by linearly interpolating the cumulative sums of ordered Lyapunov exponents to estimate the dimension of a chaotic attractor. It then distinguishes this Lyapunov-spectrum quantity from the information dimension of the invariant measure, which characterizes how the invariant measure fills phase space at increasingly fine resolutions.

The equality between these two dimensions is identified as the Kaplan–Yorke conjecture. The paper states that the conjecture has been proved only in special cases, leaving its validity in general systems unresolved. This conjecture is presented as background rather than as a problem solved by the paper, whose main contribution concerns the Lyapunov spectrum of a random neural network.

References

Kaplan and Yorke conjectured that $D_{\mathrm{KY}$ equals the information dimension of the invariant measure, which describes how the attractor fills phase space at increasingly fine resolution. This conjecture has been proven in special cases.

— Lyapunov spectrum of random neural networks  (2610.12426 - Clark, 8 Oct 2026) in Section 1, subsection “Attractor dimension and entropy rate”