Different forms of entropy dimension for zero entropy systems.

The aim of this paper is to introduce the lowers-topological entropy to distinguish zero entropy systems. That this quantity is an invariant factor under topological conjugacy and a power rule is shown. Some examples are given to show that the lower entropy dimension can attain any value in (0, 1), and are different with the upper one and the entropy dimension in the sense of Bowen. A counterexample is used to indicate that the product rule does not hold, and the lowers-topological entropy of the subsystem for the non-wandering set can be strictly less than that of the system when 0 <s< 1. Finally, this study also constructs a dynamical system to show that the transitive system with zero entropy dimension may not be minimal. [ABSTRACT FROM PUBLISHER]