Preimage entropy dimension of topological dynamical systems
  We propose a new definition of preimage entropy dimension for continuous maps on compact metric spaces, investigate fundamental properties of the preimage entropy dimension, and compare the preimage entropy dimension with the topological entropy dimension. The defined preimage entropy dimension holds various basic properties of topological entropy dimension, for example, the preimage entropy dimension of a subsystem is bounded by that of the original system and topologically conjugated systems have the same preimage entropy dimension. Also, we discuss the relation between the preimage entropy dimension and the preimage entropy. 
Lei Liu and Xiaomin Zhou and Xiaoyao Zhou
math.DS