Generic periodic and full-support measures for upper Lyapunov exponents

Let $M$ be a $d$-dimensional compact Riemannian manifold, and let $T$ be an expanding self-map of $M$. For an ergodic invariant measure, let $\lambda_1$ denote its upper Lyapunov exponent, and call a measure maximizing when

$$\lambda_1(\mu)=\sup_{\nu\in\mathcal{M}_{\mathrm{inv}}(T)}\lambda_1(\nu).$$

Upper Lyapunov exponent maximizing-measure conjecture. If $k\geq 2$, then for a $C^k$-generic expanding self-map $T$, the upper Lyapunov exponent has a unique maximizing measure, supported on a periodic orbit of $T$. If $k=1$, then for a $C^1$-generic expanding self-map $T$, the upper Lyapunov exponent has a unique maximizing measure, with zero entropy and full support.

The statement proposes a higher-dimensional analogue of generic periodic Lyapunov optimization. The paper records it as an open conjecture; the contrasting $C^1$ conclusion reflects the full-support phenomenon known for minimizing measures in the one-dimensional setting.