On Fractal Properties for Pre-image Entropy
Teng-San Shih *
Department of Applied Mathematics, Chinese Culture University, Yangmingshan, Taipei, 11114, Taiwan.
*Author to whom correspondence should be addressed.
Abstract
Fractal dimension for pre-image entropy is introduced for continuous maps throughout this paper. First we show the definition of pre-image entropy dimension of a dynamical system from different topological versions. Then we give those basic propositions of pre-image entropy dimension and the formula for power inequality and forward generator. Relationships among different types of pre-image entropy dimension are studied and an inequality relating them is given. Some basic examples are provided to compare those values of polynomial growth type with the pre-image entropy dimension. After that, this study constructs a symbolic subspace to attain any value between 0 and 1 for pre-image entropy dimension.
Keywords: Fractal dimension, forward generator, polynomial growth type