Uniform sets and complexity

Abstract

For a nonempty closed subset Ω of { 0 , 1 } Σ , where Σ is a countably infinite set, let p Ω ( S ) ≔ # π S Ω be the complexity function depending on the nonempty finite sets S ⊂ Σ , where # denotes the number of elements in a set and π S : { 0 , 1 } Σ → { 0 , 1 } S is the projection. Define the maximal pattern complexity function p Ω ∗ ( k ) ≔ sup S ; # S = k p Ω ( S ) as a function of k = 1 , 2 , … . We call Ω a uniform set if p Ω ( S ) depends only on # S = k , and the complexity function p Ω ( k ) ≔ p Ω ( S ) as a function of k = 1 , 2 , … is called the uniform complexity function of Ω . Of course, we have p Ω ( k ) = p Ω ∗ ( k ) in this case. Such uniform sets appear, for example, as the partitions generated by congruent sets in a space with optimal positionings, or they appear as the restrictions of a symbolic system to optimal windows. Let Ω ′ be the derived set (i.e. the set of accumulating points) of Ω and deg Ω ≔ inf { d ; Ω ( d + 1 ) = 0̸ } with Ω ( 1 ) = Ω ′ , Ω ( 2 ) = ( Ω ′ ) ′ , … . We prove that for any nonempty closed subset Ω of { 0 , 1 } N , where N = { 0 , 1 , 2 , … } , such that deg ( Ω ∘ ρ ) < ∞ for some injection ρ : N → N , there exists an increasing injection ϕ : N → N such that Ω ∘ ϕ ∘ ψ = Ω ∘ ϕ for any increasing injection ψ : N → N . Such a set Ω ∘ ϕ is called a super-stationary set. Moreover, if deg ( Ω ∘ ρ ) = ∞ for any injection ρ : N → N , then p Ω ∗ ( k ) = 2 k ( k = 1 , 2 , … ) holds. A uniform set Ω ⊂ { 0 , 1 } Σ is said to have a primitive factor [ Ω ∘ ϕ ] if there exists an injection ϕ : N → Σ such that Ω ∘ ϕ is a super-stationary set, where [ Ω ∘ ϕ ] is the isomorphic class containing Ω ∘ ϕ . Then, any uniform set has at least one primitive factor, and hence, any uniform complexity function is realized by the uniform complexity function of a super-stationary set. It follows that the uniform complexity function p Ω ( k ) is either 2 k for any k or a polynomial function of k for large k .