Extension to multiple negative Lyapunov exponents

Determine whether the exact dimensionality and entropy–dimension result for stationary measures of $C^1$ random diffeomorphisms with an arbitrary driving measure can be extended to systems having several distinct negative Lyapunov exponents.

Background

The paper proves exact dimensionality for ergodic stationary measures of random C1C^1 diffeomorphisms when all Lyapunov exponents collapse to a single negative value. Under a logarithmic C1C^1 moment condition and an arbitrary Borel driving measure, the stationary measure has dimension equal to the Furstenberg entropy divided by the absolute value of the common Lyapunov exponent.

The discussion contrasts this single-scale result with entropy–dimension formulas for systems having multiple Lyapunov exponents, where distinct exponential scales generally require additional geometric structures or filtrations. The unresolved question is whether the paper’s low-regularity, arbitrary-driving-measure framework can be extended beyond the single negative Lyapunov scale to several distinct negative exponents.

References

Can finite-resolution estimates and suitable nonconcentration or entropy hypotheses give summable $L1$ smoothing differences for stationary measures or their projections?

— Absolute continuity and dimension conservation for self-similar sets and measures  (2609.29301 - Jin et al., 24 Sep 2026) in Section 7, “Prospects and open questions,” subsection “Stationary measures and their projections”

It is therefore natural to ask whether the $C1$, arbitrary driving measure result proved here can be extended to several distinct negative Lyapunov exponents.

— Exact dimensionality of stationary measures for nonuniformly conformally contracting random diffeomorphisms  (2609.10538 - Mukherjee, 9 Sep 2026) in Section 1, subsection “Discussion”