Infinitesimal geometry and superstationary factors of dynamical systems

Let A be a finite set with # A ⩾ 2 and N = { 0 , 1 , 2 , … } . A nonempty closed set Θ ⊂ A N is called a superstationary set if for any infinite set { N 0 < N 1 < ⋯ } ⊂ N , we have { ω ( N 0 ) ω ( N 1 ) ⋯ ; ω ∈ Θ } = Θ . That is, Θ remains invariant for any selective observations, say at N 0 < N 1 < ⋯  . enerally, let Σ be any infinite set and Ω ⊂ A Σ be a nonempty set. Let χ be a nonprincipal ultrafilter on Σ and let Ω [ χ ∞ ] be the projective limit of Ω [ χ k ] , where Ω [ χ k ] is the value at the product ultrafilter χ k of the natural extension of the mapping S ↦ Ω [ S ] from S = ( s 1 , … , s k ) ∈ Σ k to a subset of A k given by Ω [ S ] = { ω ( s 1 ) ⋯ ω ( s k ) ; ω ∈ Ω } . We prove that Ω [ χ ∞ ] is a superstationary set, which we call a superstationary factor of Ω at χ. study of dynamical systems with time parameter, quantities which are sensitive to the time scaling, such as entropy have been of exclusive interest. On the contrary, superstationary factors represent properties depending only on time order, and not on the spacing of time. These properties are shown to reflect local aspects of the geometry behind it. o discuss a stronger and constructive version of superstationary factors.