Abstract:

Let X be a compact metric space, T:X→X be a continuous transformation, and m be a Borel measure on X. The mean topological entropy H^* (T,m) and mean measure theoretical entropy H_*(T,m) of T respect to m are defined via the localization of topological entropy and measure theoretical entropy of T. H^*(T,m) (resp. H_*(T,m)) is the weight of topological (resp. Measure theoretical) entropies of corresponding m topological (resp. Measure theoretical) chaotic attractors. So H^*(T,m) (resp. H_*(T,m)) is positive if and only if T has an  m topological (resp. measuretheoretical) chaotic attractor. For interval map f:I→I, the mean topological entropy repect to Lebesgue measure of f is denoted by H(f). It is proved that both {f:I→I: H(f)>c} and {f:I→I: H(f)=0} are dense in C⁰(I,I).

Key words: 　Ｍｅａｎｔｏｐｏｌｏｇｉｃａｌｅｎｔｒｏｐｙ, Ｍｅａｎ ｍｅａｓｕｒｅｔｈｅｏｒｅｔｉｃｅｎｔｒｏｐｙ, m-ａｔｔｒａｃｔｏｒ, m-
ｔｏｐｏｌｏｇｉｃａｌｃｈａｏｔｉｃａｔｔｒａｃｔｏｒ, m-ｍｅａｓｕｒｅｔｈｅｏｒｅｔｉｃｃｈａｏｔｉｃａｔｔｒａｃｔｏｒ．