Endomorphisms, train track maps, and fully irreducible monodromies

Irreducible Nonsurjective Endomorphisms of $F_n$ are Hyperbolic

Previously, Reynolds showed that any irreducible nonsurjective endomorphism can be represented by an irreducible immersion on a finite graph. We give a new proof of this and also show a partial converse holds when the immersion has connected Whitehead graphs with no cut vertices. The next result is a characterization of finitely generated subgroups of the free group that are invariant under an irreducible nonsurjective endomorphism. Consequently, irreducible nonsurjective endomorphisms are fully irreducible. The characterization and Reynolds' theorem imply that the mapping torus of an irreduci…

$F_n$的不可约自同态的动力学

Dynamics of Irreducible Endomorphisms of $F_n$

考虑自由群$F_n$的类非满射不可约自同态。我们发现，这样的自同态$ϕ$是拓扑表示的标记图$G$的单纯浸入$f:G \rightarrow G$;沿着我们分类的动态$\partial ϕ$作用于$\partial F_n$：有最多$2n$不动点，所有这些都是吸引。在对$ϕ$施加了一个必要的附加假设之后，我们考虑了$ϕ$对Culler-Vogtmann外层空间闭合$\bar{CV}_n$的作用。我们发现，$ϕ$作用于$\bar{CV}_n$与“汇”的动力学：有一个唯一的不动点$[T_ϕ]$，这是吸引;对于任何紧凑的邻域$N$的$[T_ϕ]$，有$K=K(N)$，使$\bar{CV}_nϕ^{K(N)} \subseteq N$。证明使用来自不变长度措施的树的certian投影。这些想法被扩展到显示如何分解树$T$在外层空间的边界考虑$T$不变长度措施的空间，这给出了一个分解，推广了分解的几何树来自Imanishi的定理。

Irreducibility of a Free Group Endomorphism is a Mapping Torus Invariant

We prove that the property of a free group endomorphism being irreducible is a group invariant of the ascending HNN extension it defines. This answers a question posed by Dowdall-Kapovich-Leininger. We further prove that being irreducible and atoroidal is a commensurability invariant. The invariance follows from an algebraic characterization of ascending HNN extensions that determines exactly when their defining endomorphisms are irreducible and atoroidal; specifically, we show that the endomorphism is irreducible and atoroidal if and only if the ascending HNN extension has no infinite index s…

有限生成自由半群自同态的不可约性

Irreducibility of Endomorphisms of Finitely Generated Free Semigroups

本文研究有限生成自由半群自同态的不可约性，引入素性概念，给出保秩自同态可约与不可约的判别条件，并分析分解的非唯一性及关联矩阵的应用。

强不可约子移位中周期组态的稠密性

On the density of periodic configurations in strongly irreducible subshifts

设$G$是剩余有限群，$A$是有限集.证明了如果$X \subset A^G$是包含周期组态的有限型强不可约子移位，则周期组态在$X$中是稠密的.周期配置的密度特别意味着$X$的每个内射自同态是满射的，并且$X$的自同构群是剩余有限的。我们还引入了一类子移位$X \subset A^\Z$，包括所有强不可约子移位和所有不可约sofic子移位，其中周期组态是稠密的。

Hyperbolic Immersions of Free Groups

We prove that the mapping torus of a graph immersion has a word-hyperbolic fundamental group if and only if the corresponding endomorphism does not produce Baumslag-Solitar subgroups. Due to a result by Reynolds, this theorem applies to all injective endomorphisms of $F_2$ and nonsurjective fully irreducible endomorphisms of $F_n$. We also give a framework for extending the theorem to all injective endomorphisms of $F_n$.