A New Version of Distributional Chaos, Distributional Chaos in a Sequence, and Other Concepts of Chaos

Abstract

In this paper, we investigate the relations between distributional chaos in a sequence and distributional chaos ([Formula: see text]-chaos, R–T chaos, DC3, respectively). Firstly, we prove a sufficient condition that the distributional chaos is equivalent to the distributional chaos in a sequence. Besides, we prove that distributional chaos in a sequence and [Formula: see text]-chaos (R–T chaos, DC3, respectively) do not imply each other. Finally, we give a new definition of chaos, named DC2[Formula: see text], which is similar to DC2, and show that DC2[Formula: see text] is an invariant of topological conjugacy and an iteration invariant (that is, for any integer [Formula: see text], [Formula: see text] is DC2[Formula: see text] if and only if [Formula: see text] is DC2[Formula: see text]).