Local rigidity of measure and Lyapunov exponents

Let $\alpha$ be a $C^2$ action with an invariant measure $\mu$ satisfying the assumptions of Main Theorem (1) or (2). For an action $\alpha'$ sufficiently close to $\alpha$ in the $C^2$ topology, consider invariant measures $\mu'$ and weak* convergence of measures.

Local measure-rigidity conjecture. The action $\alpha'$ has an ergodic invariant measure $\mu'$ satisfying the same assumptions; $\mu'$ can be chosen so that $\mu'\to\mu$ in the weak* topology whenever $\alpha'\to\alpha$ in the $C^2$ topology. In the $\mathbb Z^k$ case, the Lyapunov exponents of $\mu'$ equal those of $\mu$.

This is proposed as a local counterpart of global measure rigidity, motivated by the continuous variation of the distinguished invariant measure in the toral Cartan setting. The paper does not establish it.