Topological entropy of the set of points without physical-like behaviour and away from measures satisfying the entropy formula beyond uniform hyperbolicity

We study the topological entropy of the set of points without physical-like behaviour and away from measures satisfying the entropy formula in differentiable dynamical systems beyond uniform hyperbolicity. On the one hand, for partially hyperbolic systems with multiple one-dimensional centers, we prove that the topological entropy of this set is greater than or equal to the supremum of the metric entropy of all hyperbolic measures. Additionally, for Ma nédiffeomorphisms, we show that the topological entropy of this set is equal to the topological entropy of the entire system. On the other hand, for differentiable dynamical systems where the number of ergodic physical measures and ergodic measures satisfying the entropy formula is countable and each ergodic physical measure and ergodic measure satisfying the entropy formula is hyperbolic, we prove that the topological entropy of this set is equal to the topological entropy of the entire system, and we present applications of this result. Finally, we prove that for $C^1$-generic volume-preserving diffeomorphisms, the topological entropy of this set is greater than or equal to the metric entropy of the volume measure.