A characterization of subshifts with bounded powers

Arnoux-Rauzy Subshifts: Linear Recurrence, Powers, and Palindromes

We consider Arnoux-Rauzy subshifts $X$ and study various combinatorial questions: When is $X$ linearly recurrent? What is the maximal power occurring in $X$? What is the number of palindromes of a given length occurring in $X$? We present applications of our combinatorial results to the spectral theory of discrete one-dimensional Schrödinger operators with potentials given by Arnoux-Rauzy sequences.

Abelian Complexity in Minimal Subshifts

In this paper we undertake the general study of the Abelian complexity of an infinite word on a finite alphabet. We investigate both similarities and differences between the Abelian complexity and the usual subword complexity. While the Thue-Morse minimal subshift is neither characterized by its Abelian complexity nor by its subword complexity alone, we show that the subshift is completely characterized by the two complexity functions together. We give an affirmative answer to an old question of G. Rauzy by exhibiting a class of words whose Abelian complexity is everywhere equal to 3. We also …

拉马努金子移位

Ramanujan subshifts

本文提出了一种高维拉马努金子移位的构造方法，通过四元数格生成非二分的(q+1)-正则拉马努金图，并提供了生成无限家族的提升规则。

Ends of Schreier graphs and cut-points of limit spaces of self-similar groups

Every self-similar group acts on the space $X^ω$ of infinite words over some alphabet $X$. We study the Schreier graphs $Γ_w$ for $w\in X^ω$ of the action of self-similar groups generated by bounded automata on the space $X^ω$. Using sofic subshifts we determine the number of ends for every Schreier graph $Γ_w$. Almost all Schreier graphs $Γ_w$ with respect to the uniform measure on $X^ω$ have one or two ends, and we characterize bounded automata whose Schreier graphs have two ends almost surely. The connection with (local) cut-points of limit spaces of self-similar groups is established.

The size-Ramsey number of powers of bounded degree trees

Given a positive integer $s$, the $s$-colour size-Ramsey number of a graph $H$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that, in any colouring of $E(G)$ with $s$ colours, there is a monochromatic copy of $H$. We prove that, for any positive integers $k$ and $s$, the $s$-colour size-Ramsey number of the $k$th power of any $n$-vertex bounded degree tree is linear in $n$. As a corollary we obtain that the $s$-colour size-Ramsey number of $n$-vertex graphs with bounded treewidth and bounded degree is linear in $n$, which answers a question rai…

边理想的有界幂：对称交换二项式

Bounded powers of edge ideals: symmetric exchange binomials

本文研究离散多拟阵基环的toric理想生成问题，证明了几类离散多拟阵（特别是源于有限图边理想有界幂的）的toric理想可由对称交换二项式生成。