Distributional Chaos in the Zero Topological Entropy Subsets of Non-Dense Orbits

In this paper, we mainly focus on the set of non-dense points of a dynamical system. We study the distributional chaos in such set. As for a mixing expanding map or a transitive Anosov diffeomorphism on a compact connected manifold, we prove that DC1 chaos can occur in a zero topological entropy subset of the intersection of the set of recurrent points and the set of the non-dense points. Also, for such dynamical systems, strongly distributional chaos (which is stronger than DC1 chaos) can occur in a zero topological entropy subset of the set of non-recurrent points. Besides, when we divide the total space into six layers according to the different statistical structures, similar results can appear in every layer. Our results can also be applied to mixing subshifts of finite type, \(\beta \)-shifts, homoclinic classes and \(C^{1+\alpha }\) diffeomorphisms preserving a weakly mixing hyperbolic ergodic measure.