The weak non-uniform hyperbolicity conjecture for physical measures

Let $f$ be a diffeomorphism and let the attracting set under consideration have a dominated splitting. Weak non-uniform expansion means that

$$\liminf_{n\nearrow\infty} S_n\phi^{cu}(x)/n<0$$

on a positive-volume subset of points in the trapping region, and weak non-uniform contraction means that

$$\liminf_{n\nearrow\infty} S_n\phi^{cs}(x)/n<0$$

on such a subset. Weak non-uniform hyperbolicity conjecture. Every attracting set with a dominated splitting with both weak non-uniform expansion and weak non-uniform contraction admits a physical measure. This would extend existing results producing physical measures from weak non-uniform expansion or contraction in partially hyperbolic settings, and would allow the paper's mixing theorem to apply to the resulting physical measure.