Kazutoyo Iketake | Searcharxiv

arXiv2026

Galois Theory for Subshifts of Finite Type and Representations of Automorphism Groups

The purpose of this paper is to constructively develop a Galois theory on irreducible shifts of finite type (SFTs) and to analyze the automorphism groups of SFTs using this framework. Let $X$ and $Y$ be irreducible SFTs. We demonstrate that a factor map $Y \to X$ satisfying algebraic conditions exhibits Galois-theoretic behavior. Specifically, we prove that a Galois correspondence, analogous to those known in the Galois theory of fields and covering spaces in topology, holds for irreducible SFTs. Furthermore, we introduce the absolute Galois group and its cohomology as conjugacy invariants for irreducible SFTs. Finally, we construct a representation of the automorphism group of an irreducible SFT into this cohomology to extract information from the automorphism group, whose structure is not yet fully understood.

math.DS↗

arXiv2025

Existence and Configuration of Invariant Sets in $C^\infty([a,b])$ on which the Differential Operator Exhibits Devaney's Chaos

In this paper, we investigate the chaotic behavior of the differential operator $\frac{d}{dx}$ on the space of smooth functions $C^\infty([a,b])$ equipped with the $L^p$-norm ($1\le p\le\infty$). We explicitly construct a homeomorphism between a subset of $C^\infty([a,b])$ and the shift space. Moreover, inspired by symbolic dynamics, we demonstrate that invariant sets, on which the differential operator behaves analogously to the shift, are densely configured in $C^\infty([a,b])$. We also prove that the differential operator is chaotic on the entire space $C^\infty([a,b])$ using a similar approach.

math.DS↗