Existence and uniqueness conjecture for minimizing upper-Lyapunov measures

Let $M$ be a manifold and let $T$ be an expanding self-map. For an invariant measure $\mu$, let $\lambda_1(\mu)$ denote the upper Lyapunov exponent, and call $\mu$ minimizing when

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

Minimizing upper-Lyapunov measure conjecture. For a generic expanding self-map $T$ on a manifold $M$, the minimizing measure of the upper Lyapunov exponent $\lambda_1$ exists, is unique, and has zero entropy.

The paper notes that existence of a minimizing measure for the upper Lyapunov exponent is not known in general, which motivates this conjecture. It does not specify a topology or regularity class in the formulation.