On the specification property and synchronisation of unique $q$-expansions

Beyond Bowen's Specification Property

A classical result in thermodynamic formalism is that for uniformly hyperbolic systems, every Hölder continuous potential has a unique equilibrium state. One proof of this fact is due to Rufus Bowen and uses the fact that such systems satisfy expansivity and specification properties. In these notes, we survey recent progress that uses generalizations of these properties to extend Bowen's arguments beyond uniform hyperbolicity, including applications to partially hyperbolic systems and geodesic flows beyond negative curvature. We include a new criterion for uniqueness of equilibrium states for …

On intrinsic ergodicity and weakenings of the specification property

Since seminal work of Bowen, it has been known that the specification property implies various useful properties about a topological dynamical system, among them uniqueness of the measure of maximal entropy (often referred to as intrinsic ergodicity). Weakenings of the specification property called almost weak specification and almost specification have been defined and profitably applied in various works. However, it has been an open question whether either or both of these properties imply intrinsic ergodicity. We answer this question negatively by exhibiting examples of subshifts with multi…

Specification and towers in shift spaces

We show that a shift space on a finite alphabet with a non-uniform specification property can be modeled by a strongly positive recurrent countable-state Markov shift to which every equilibrium state lifts. In addition to uniqueness of the equilibrium state, this gives strong statistical properties including the Bernoulli property, exponential decay of correlations, central limit theorem, and analyticity of pressure, which are new even for uniform specification. We give applications to shifts of quasi-finite type, synchronised and coded shifts, and factors of beta-shifts and S-gap shifts.

The weak specification property for geodesic flows on CAT(-1) spaces

We prove that the geodesic flow on a compact locally CAT(-1) space has the weak specification property, and give various applications. We show that every Hölder potential on the space of geodesics has a unique equilibrium state. We establish the equidistribution of weighted periodic orbits and the large deviations principle for all such measures. The thermodynamic results are proved for the class of expansive flows with weak specification.

具有扩张性和规范性的非紧系统的热力学形式

Thermodynamic formalism for non-compact systems with expansivity and specification

针对完备可分度量空间上的连续流，利用规范性质发展了平衡态理论，引入强正递归概念证明平衡态的存在唯一性，应用于无挤压负曲率流形和CAT(-1)空间上的测地流。

Intrinsic ergodicity via obstruction entropies

Bowen showed that a continuous expansive map with specification has a unique measure of maximal entropy. We show that the conclusion remains true under weaker non-uniform versions of these hypotheses. To this end, we introduce the notions of obstructions to expansivity and specification, and show that if the entropy of such obstructions is smaller than the topological entropy of the map, then there is a unique measure of maximal entropy.

Unique equilibrium states for flows and homeomorphisms with non-uniform structure

Using an approach due to Bowen, Franco showed that continuous expansive flows with specification have unique equilibrium states for potentials with the Bowen property. We show that this conclusion remains true using weaker non-uniform versions of specification, expansivity, and the Bowen property. We also establish a corresponding result for homeomorphisms. In the homeomorphism case, we obtain the upper bound from the level-2 large deviations principle for the unique equilibrium state. The theory presented in this paper provides the basis for an ongoing program to develop the thermodynamic for…

唯一伯努利吉布斯态与g-测度

Unique Bernoulli Gibbs states and g-measures

本文给出平移不变规范场的吉布斯态成为g-链的充分条件，并据此推导出唯一且平移不变、伯努利的吉布斯态存在的判据。

On intrinsic ergodicity of factors of $\mathbb{Z}^d$ subshifts

It is well-known that any $\mathbb{Z}$ subshift with the specification property has the property that every factor is intrinsically ergodic, i.e., every factor has a unique factor of maximal entropy. In recent work, other $\mathbb{Z}$ subshifts have been shown to possess this property as well, including $β$-shifts and a class of $S$-gap shifts. We give two results that show that the situation for $\mathbb{Z}^d$ subshifts with $d >1 $ is quite different. First, for any $d>1$, we show that any $\mathbb{Z}^d$ subshift possessing a certain mixing property must have a factor with positive entropy w…

Weighted equilibrium states for factor maps between subshifts

Let $π:X\to Y$ be a factor map, where $(X,σ_X)$ and $(Y,σ_Y)$ are subshifts over finite alphabets. Assume that $X$ satisfies weak specification. Let $\ba=(a_1,a_2)\in \R^2$ with $a_1>0$ and $a_2\geq 0$. Let $f$ be a continuous function on $X$ with sufficient regularity (Hölder continuity, for instance). We show that there is a unique shift invariant measure $μ$ on $X$ that maximizes $μ(f)+a_1h_μ(σ_X)+ a_2h_{μ\circ π^{-1}}(σ_Y)$. In particular, taking $f\equiv 0$ we see that there is a unique invariant measure $μ$ on $X$ that maximizes the weighted entropy $a_1h_μ(σ_X)+ a_2h_{μ\circ π^{-1}}(σ_Y…

一类斜积变换的不规则集

Irregular sets for a class of skew product transformations

针对由唯一遍历同胚系统驱动且纤维满足Anosov与拓扑混合性质的斜积变换，本文证明若某纤维上连续可观测量的不规则集非空，则$\boldsymbol{\text{P}}$-几乎处处纤维上的该集合必非空、剩余且具全纤维拓扑熵，关键技术为纤维规格性质的轨道粘合。