【115/10/8】16:20-17:10 張志鴻 特聘教授 (國立高雄大學應用數學系)
發佈日期 :
2026-10-02
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| Time | 2026-10-8 16:20-17:10 |
| Venue | 數學館1樓 31106 教室 |
| Speaker | 張志鴻 特聘教授 (國立高雄大學應用數學系) |
| Title | Nonstationary Shifts of Finite Type with Controllers |
| Abstract | Let A and B be finite alphabets, let S be a shift space over B, and let A = {Aᵢ} (i ∈ B) be a family of binary matrices indexed by A. We study the nonstationary shift of finite type with controller S, X(S,A) = {x ∈ A^Z : there exists s ∈ S such that Aₛᵢ(xᵢ, xᵢ₊₁) = 1 for all i ∈ Z}, in which the symbol sᵢ selects the transition matrix governing the passage from xᵢ to xᵢ₊₁. Letting an arbitrary, possibly nonsofic, controller drive the switching keeps the Arnoux–Fisher construction shift-invariant. The consideration extends to nonstationary shift spaces with finite memory X(S,X)^(m). After showing that every X(S,X)^(m) is conjugate to some X(S,A), we investigate X(S,A) from the viewpoint of controllers. For sofic controllers we construct an explicit finite product presentation, derive the entropy formula h(X(S,A)) = log ρ(M) from a right-resolving output presentation, and characterize positive entropy by a branching condition on A and L(S); for nonsofic controllers the construction persists as an infinite follower-set presentation. This talk concerns the irreducibility and mixing of NSFTs. Beyond the sufficient condition furnished by a positive matrix word A_w > 0, we introduce joint conditions on (S,A)—positivity of the Boolean joins of the products A_z over all controller words z insertable between prescribed elements of L(S)—and prove that they characterize irreducibility, mixing and the specification property of the controlled extension Y(S,A), are sufficient for X(S,A), and are necessary for X(S,A) whenever A is unambiguous. All three conditions are decidable for sofic controllers. Sensitivity, multi-sensitivity and Li–Yorke chaos follow as consequences. |
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