Ergodicity of extremal invariant measures for Brody curves

Determine whether, for each real number c with 0 < c < 2(N+1)ρ(ℂP^N), there exists an ergodic T-invariant Borel probability measure μ on the space of Brody curves 𝔅^N such that rdim(𝔅^N, T, d, μ) = ∫_{𝔅^N} ψ dμ = c; in particular, ascertain whether there exists at least one ergodic measure ν on 𝔅^N with rdim(𝔅^N, T, d, ν) = ∫_{𝔅^N} ψ dν > 0.

Background

Theorem 1.2 in the paper constructs many invariant measures μ on the space of Brody curves that attain equality in the Ruelle-type inequality: rdim(𝔅^N, T, d, μ) = ∫_{𝔅^N} ψ dμ, for any target value c below the supremum. However, the construction does not clarify ergodic properties of these measures, such as whether any of them are ergodic or mixing.

Using ergodic decomposition, the author shows that upper rate distortion dimension equality holds almost surely for components, but this does not yield control over lower rate distortion dimension, leaving the existence of ergodic measures with the desired equality and positive value unresolved.