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In this article we prove that for a C 1+α diffeomorphism on a compact Riemannian manifold, if there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then the topological entropy of the set of irregular points that are not uniformly hyperbolic is larger than or equal to the metric entropy of the hyperbolic ergodic measure. In the process of proof, we give an abstract general mechanism to study topological entropy of irregular points provided that the system has a sequence of nondecreasing invariant compact subsets such that every subsystem has shadowing property and is transitive.Theorem 1.1. [16, Corollary B] Let M be a compact Riemannian manifold of dimension at least 2 and f be a C 1+α diffeomorphism. Then one has h top (f, IR(f )) ≥ sup{h top (f, Λ) : Λ is a transitive locally maximal hyperbolic set} = sup{h µ (f ) : µ is a hyperbolic ergodic measure}.

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In this article, we prove that the set of points whose pointwise emergence have same degree in [ 1 , ∞ ] with repect to polynomial has strong dynamical complexity in sense of entropy, residuality and density, and distributional chaos for a β-shift, a C 1 diffeomorphism having a nontrival homoclinic class, or a C 1 + α diffeomorphism preserving a hyperbolic ergodic measure. In this process, we construct nonempty compact connected set with fixed box-counting dimension in the space of invariant measures. Then combining with the relation between pointwise emergence of saturated set and box-counting dimension, we get the goal from two frameworks about dynamical complexity of saturated set. One framework was introduced by C. Pfister and W. Sullivan and can be applicable to transitive Anosov diffeomorphisms, mixing expanding maps, β-shifts, transitive subshifts of finite type, mixing sofic subshifts, and mixing S-gap shifts. The other framework is applicable to general homoclinic classes and diffeomorphisms preserving hyperbolic ergodic measures. We also get some combined observations with irregular set.

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