Generic existence of regular points and Birkhoff averages

Let a dynamical system have a positive-volume subset of its ambient space. A point is $p\omega$-regular if it has the regularity property denoted by $p\omega$ in the source; forward Birkhoff averages and forward Lyapunov exponents are understood in the usual sense. Generic regularity conjecture. For a dense/generic subset of dynamical systems, there exists a positive-volume subset in the ambient space such that at least one of the following holds: the points are $p\omega$-regular; forward Birkhoff averages exist at all points; or forward Lyapunov exponents exist and the points are Lyapunov regular. This would yield physical measures for the smooth systems in the open classes of non-uniformly hyperbolic flows and diffeomorphisms discussed in the paper, and a criterion for obtaining physical/SRB measures from the sectional Lyapunov spectrum.