A characterization of physical measures for systems with mixed central behavior

We show that the existence of physical measures for $C^{\mathrm{\infty }}$ smooth instances of certain partially hyperbolic dynamics, both continuous and discrete, exhibiting mixed behavior (positive and negative Lyapunov exponents) along the central non-uniformly hyperbolic multidimensional invariant direction, is equivalent to the existence of certain types of ``regular points’’ on positive volume subsets, including Lyapunov regular points. This encompasses the $C^3$ robust class of multidimensional non-hyperbolic attractors obtained by Viana, and the $C^1$ robust classes of $3$-sectionally hyperbolic wild strange attractors presented by Shilnikov and Turaev, providing necessary and sufficient conditions for the existence of ergodic hyperbolic physical measures on these and other dynamical systems.