Behavior of various complexity functions

For a nonempty closed set Ω ⊂ AN with 2 ≤ #A < ∞, we consider 3 complexity functions of k = 0, 1, 2, · · · : (1) (block complexity) p Ω (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.