Behavior of various complexity functions

AbstractFor a nonempty closed set Ω⊂AN with 2≤#A<∞, we consider 3 complexity functions of k=0,1,2,…: (1)(block complexity) pΩBL(k):=#Ω|{0,1,…,k−1},(2)(maximal pattern complexity) pΩ∗(k):=supS⊂N,#S=k#Ω|S,(3)(minimal pattern complexity) p∗Ω(k):=infS⊂N,#S=k#Ω|S, where # denotes the number of elements in a set, and Ω|S is the restriction of Ω to S⊂N. If pΩ∗(k)=p∗Ω(k)(k=0,1,2,…) holds, then the above 3 complexities coincide and are called uniform complexity, denoted by pΩ(k).Behaviors of these 4 complexity functions are discussed