On the Perron root and eigenvectors associated with a subshift of finite type

本原替换的子移位的几何表示

Geometrical representation of subshifts for primitive substitutions

本文研究 Perron 特征值为 Pisot 单位的本原替换的子移位，构造了可测共轭的域交换，给出环面平移有限扩张条件，证明弱不可约 Pisot 替换子移位性质，并提供计算伪幺模替换子移位特征值的算法。

关于 (A, B, P, ϕ) 平衡双线性模型的 Perron-Frobenius 强阈值定理，以及左、右 Perron 特征向量在数学流行病学中的作用

A Perron-Frobenius strong threshold theorem for (A, B, P, ϕ) balanced bilinear models, and the role of left and right Perron eigenvectors in mathematical epidemiology

本文扩展了秩一 NGM 双线性模型的阈值定理，证明了非平凡正平衡存在的充要条件，并利用左、右 Perron 特征向量构建 Lyapunov 函数，探讨了非秩一情况下的推广。

A combinatorial approach to study subshifts associated with multigraphs

A subshift of finite type over finitely many symbols can be described as a collection of all infinite walks on a digraph with at most a single edge from a vertex to another. The associated finite set $\F$ of forbidden words is a constraint which determines the language of the shift entirely. In this paper, in order to describe infinite walks on a multigraph, we introduce the notion of multiplicity of a word (finite walk) and define repeated words as those having multiplicity at least $2$. In general, for given collections $\F$ of forbidden words and $\R$ of repeated words with pre-assigned mul…

A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors

We establish a generalized Perron-Frobenius theorem, based on a combinatorial criterion which entails the existence of an eigenvector for any nonlinear order-preserving and positively homogeneous map $f$ acting on the open orthant $\mathbb{R}_{\scriptscriptstyle >0}^n$. This criterion involves dominions, i.e., sets of states that can be made invariant by one player in a two-person game that only depends on the behavior of $f$ "at infinity". In this way, we characterize the situation in which for all $α, β> 0$, the "slice space" $\mathcal{S}_α^β:= \{ x \in \mathbb{R}_{\scriptscriptstyle >0}^n \…

A Perron-Frobenius type result for integer maps and applications

It is shown that for certain maps, including concave maps, on the $d$-dimensional lattice of positive integer points, 'approximate' eigenvectors can be found. Applications in epidemiology as well as distributed resource allocation are discussed as examples.

Shifts of finite type and random substitutions

We prove that every topologically transitive shift of finite type in one dimension is topologically conjugate to a subshift arising from a primitive random substitution on a finite alphabet. As a result, we show that the set of values of topological entropy which can be attained by random substitution subshifts contains all Perron numbers and so is dense in the positive real numbers. We also provide an independent proof of this density statement using elementary methods.