Structural implications of the $d_{\mathrm{An}}$-Cauchy condition

Let $M$ be the underlying manifold, let $\theta$ be a limit flow, and let $\{X_i\}$ be its approximating generators. Let $d_{\mathrm{An}}=d_{\mathrm{An},C^0}$ be the metric on generators, and suppose that $\{X_i\}$ is Cauchy in this metric.

Structural implications of the $d_{\mathrm{An}}$-Cauchy condition. The requirement that $\{X_i\}$ be $d_{\mathrm{An}}$-Cauchy ensures that the limit flow $\theta$ inherits strong structural or stability properties beyond those guaranteed by mere $C^0$ generator convergence, such as certain shadowing properties, continuous invariant foliations or laminations, or physical/SRB-like invariant measures reflecting the hyperbolicity of the approximants.

The conjecture proposes that the stronger metric control captures more hyperbolic structure than $C^0$ convergence alone. The source lists possible consequences rather than establishing any of them.