Ledrappier–Young conjecture for quasi-decay of smooth dynamical measures

Let $(X,T,\mu)$ be a smooth ergodic measure-preserving dynamical system of dimension $d$. Let $\lambda_1<\ldots<\lambda_k$ be its Lyapunov exponents, and let $d_1,\ldots,d_k$ be the corresponding Lyapunov dimensions. For each $i=1,\ldots,k$, let $\gamma_i$ denote the transverse dimension associated with the corresponding Lyapunov data. Ledrappier–Young conjecture. If

$$\gamma_i>d_i-1$$

for every $i=1,\ldots,k$, then $\mu$ is quasi-decaying.

The conjecture proposes a criterion extending the Ledrappier–Young relationship between entropy, Lyapunov exponents, and dimension beyond the diffeomorphism setting. It is presented as implying the paper's examples for one-dimensional systems and diagonal toral endomorphisms; the source does not state a resolution.