Topological mixing for substitutions on two letters

典型Salem替换悬吊流的定量弱混合性

Quantitative weak mixing for typical Salem substitution suspension flows

本文研究了Salem替换流的定量弱混合性，证明了其弱混合性速率略逊于$\log\log$幂，并首次给出了Salem伯努利卷积的傅里叶变换的定量衰减率。

Mixing properties and entropy bounds of a family of Pisot random substitutions

We consider a two-parameter family of random substitutions and show certain combinatorial and topological properties they satisfy. We establish that they admit recognisable words at every level. As a consequence, we get that the subshifts they define are not topologically mixing. We then show that they satisfy a weaker mixing property using a numeration system arising from a sequence of lengths of inflated words. Moreover, we provide explicit bounds for the corresponding topological entropy in terms of the defining parameters $n$ and $p$.

Automorphisms of automatic shifts

In this article we continue the study of automorphism groups of constant length substitution shifts and also their topological factors. We show that up to conjugacy, all roots of the identity map are letter exchanging maps, and all other nontrivial automorphisms arise from {\em twisted} compressions of another constant length substitution. We characterise the group of roots of the identity in both the measurable and topological setting. Finally, we show that any topological factor of a constant length substitution shift is topologically conjugate to a constant length substitution shift via a l…

随机替换铺砌的定量弱混合

Quantitative weak mixing for random substitution tilings

本文研究随机替换铺砌的定量弱混合，通过估计 R^d 作用的扭曲遍历积分增长，给出谱测度下局部维数下界，并将 Bufetov-Solomyak 结果推广到高维。

The $S$-adic Pisot conjecture on two letters

We prove an extension of the well-known Pisot substitution conjecture to the $S$-adic symbolic setting on two letters. The proof relies on the use of Rauzy fractals and on the fact that strong coincidences hold in this framework.

On substitution tilings and Delone sets without finite local complexity

We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences. In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion…

Eigenfunctions for substitution tiling systems

We prove that for the uniquely ergodic ${\bf R}^d$ action associated with a primitive substitution tiling of finite local complexity, every measurable eigenfunction coincides with a continuous function almost everywhere. Thus, topological weak-mixing is equivalent to measure-theoretic weak-mixing for such actions. If the expansion map for the substitution is a pure dilation by $θ>1$ and the substitution has a fixed point, then failure of weak-mixing is equivalent to $θ$ being a Pisot number.

Factor-balanced $S$-adic languages

A set of words, also called a language, is letter-balanced if the number of occurrences of each letter only depends on the length of the word, up to a constant. Similarly, a language is factor-balanced if the difference of the number of occurrences of any given factor in words of the same length is bounded. The most prominent example of a letter-balanced but not factor-balanced language is given by the Thue-Morse sequence. We establish connections between the two notions, in particular for languages given by substitutions and, more generally, by sequences of substitutions. We show that the two…

Zeckendorf representations and mixing properties of sequences

We use generalised Zeckendorf representations of natural numbers to investigate mixing properties of symbolic dynamical systems. The systems we consider consist of bi-infinite sequences associated with so-called random substitutions. We focus on random substitutions associated with the Fibonacci, tribonacci and metallic mean numbers and take advantage of their respective numeration schemes.

Chacon映射的弱混合速度

Speed of Weak Mixing for the Chacon Map

本文通过非本原替换子移位与谱估计方法，研究区间上Chacon映射的弱混合速度，并证明所得界本质上是精确的。

Pisot替换猜想与Rauzy分形

Pisot Substitution Conjecture and Rauzy Fractals

本文利用前缀-后缀自动机导出的有限型子移位的弱混合性质，证明了Pisot猜想的一个版本，并得出替换动力系统具有纯离散谱。

Topological mixing for some residual sets of interval exchange transformations

We show that a residual set of non-degenerate IETs on more than 3 letters is topologically mixing. This shows that there exists a uniquely ergodic topologically mixing IET. This is then applied to show that some billiard flows in a fixed direction in an L-shaped polygon are topologically mixing.

超椭圆型某些线性对合的拓扑弱混合

Topological weak mixing for certain linear involutions of hyperelliptic type

研究超椭圆型某些线性对合的拓扑弱混合问题，通过考虑特定广义置换\(\sigma_{s,r}\)，证明了由其劳齐类中广义置换定义的典型线性对合的自然符号编码在有至少一个简单字母且相关亏格足够大时是拓扑弱混合的。

When size matters: subshifts and their related tiling spaces

We investigate the dynamics of substitution subshifts and their associated tiling spaces. For a given subshift, the associated tiling spaces are all homeomorphic, but their dynamical properties may differ. We give criteria for such a tiling space to be weakly mixing, and for the dynamics of two such spaces to be topologically conjugate.

Trace maps, invariants, and some of their applications

Trace maps of two-letter substitution rules are investigated with special emphasis on the underlying algebraic structure and on the existence of invariants. We illustrate the results with the generalized Fibonacci chains and show that the well-known Fricke character I(x,y,z) = x^2 + y^2 + z^2 - 2 x y z - 1 is not the only type of invariant that can occur. We discuss several physical applications to electronic spectra including the gap-labeling theorem, to kicked two-level systems, and to the classical 1D Ising model with non-commuting transfer matrices.