Irregular sets, the $β$-transformation and the almost specification property
Daniel J. Thompson
备注：
To appear in Transactions of the American Mathematical Society. Final version was submitted to TAMS on November 24th 2010, and it appears I did not update the arXiv version at the time! Changes from version 1 are mostly minor - typos fixed, references improved. The most significant change is to section 5.1, where the results have been improved
Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map satisfying a property we call almost specification (which is slightly weaker than the $g$-almost product property of Pfister and Sullivan), and $ϕ$ be a continuous function on $X$. We show that the set of points for which the Birkhoff average of $ϕ$ does not exist (which we call the irregular set) is either empty or has full topological entropy. Every $β$-shift satisfies almost specification and we show that the irregular set for any $β$-shift or $β$-transformation is either empty or has full topological entropy and Hausdorff dimension.
We show that systems with some specification properties are topologically or almost Borel universal, in the sense that any aperiodic subshift with lower entropy may be topologically or almost Borel embedded.
On entropy of dynamical systems with almost specification
构造了一族具有几乎规格和多重极大熵测度的移位空间。这回答了Climenhaga和Thompson的一个问题[Israel J. Math.192（2012），no.2，785- 817]。本文还证明了Climenhaga和Thompson给出的移位空间的每个子移位因子是内遍历的一些充分条件在某种意义上是最佳可能的，而且弱规范性质既不蕴含内遍历性，也不从几乎规范中导出.我们还构造了一个具有弱规格性质的动力系统，它不具有几乎规格性质。证明了具有几乎规格性质的系统的极小点在任意不变测度的支撑下是稠密的。此外，如果几乎规格化系统的不变测度具有非平凡支撑，则它在任一不变测度的支撑上也具有一致正熵，且不可能是最小的.
In this article we prove that for a diffeomorphism on a compact Riemannian manifold, if there is a nontrival homoclinic class that is not uniformly hyperbolic or the diffeomorphism is a $C^{1+α}$ and there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then we find a type of strongly distributional chaos which is stronger than usual distributional chaos and Li-Yorke chaos in the set of irregular orbits that are not uniformly hyperbolic. Meanwhile, we prove that various fractal sets are strongly distributional chaotic, such as irregular sets, level sets, several recu…
We study the Bowen topological entropy of generic and irregular points for certain dynamical systems. We define the topological entropy of noncompact sets for flows, analogous to Bowen's definition. We show that this entropy coincides with the Bowen topological entropy of the time-1 map on any set. We also show a Bowen's inequality for flows; namely, that the metric entropy with respect to every invariant measure for a continuous flow is an upper bound for the topological entropy of the set of generic points with respect to the same measure, and the equality is always true if the measure is er…
It follows from Oseledec Multiplicative Ergodic Theorem (or Kingmans Subadditional Ergodic Theorem) that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system f with exponential specification property and a Holder continuous matrix cocycle A, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of A displays distributional chaos of type 1.
A Borel system $(X,S)$ is `almost Borel universal' if any free Borel dynamical system $(Y,T)$ of strictly lower entropy is isomorphic to a Borel subsystem of $(X,S)$, after removing a null set. We obtain and exploit a new sufficient condition for a topological dynamical system to be almost Borel universal. We use our main result to deduce various conclusions and answer a number of questions. Along with additional results, we prove that a `generic' homeomorphism of a compact manifold of topological dimension at least two can model any ergodic transformation, that non-uniform specification impli…
Let $(X,d,f)$ be a dynamical system, where $(X,d)$ is a compact metric space and $f:X\rightarrow X$ is a continuous map. Using the concepts of \textit{g-almost product property} and \textit{uniform separation property} introduced by Pfister and Sullivan in \cite{Pfister2007}, we give a variational principle for certain non-compact with relation the asymptotically additive topological pressure. We also study the set of points that are irregular for an collection finite or infinite of asymptotically additive sequences and we show that carried the full asymptotically additive topological pressure…
本文介绍了一致空间上动力系统的规范性质（频闪性质）。我们主要研究两类动力系统：广义位移和以离散空间为相空间的Alexandroff紧化的动力系统。本文证明了对于至少含有两个元素、一个非空集$Γ$和一个自映射$φ:Γ\toΓ$的离散有限拓扑空间$X$，广义移位动力系统$(X^Γ,σ_φ)$：\begin{itemize} \item has (almost) weak specification property if and only if $φ:Γ\toΓ$ does not have any periodic point, \item has (uniform) stroboscopical property if and only if $φ:Γ\toΓ$ is one-to-one. \end{itemize}
Nonexistence of Lyapunov Exponents for Matrix Cocycles
由奥塞莱代克乘法遍历定理可知，对于给定的连续上循环的奥塞莱代克平均值发散的李雅普诺夫-非正则点集，关于任何不变的概率测度具有零测度。与此相反，对于任何具有指数规格性质的动力系统$f:X\rightarrow X$和连续矩阵上循环$A:X\rightarrow G (m,\mathbb{R})$下的H $\ddot{\text{o}}$，本文证明了：如果存在具有不同Lyapunov谱的遍历测度，则$A$的Lyapunov-非正则集是剩余的（即，包含密集的$G_δ$集合）。
We study the dynamical properties of irregular model sets and show that the translation action on their hull always admits an infinite independence set. The dynamics can therefore not be tame and the topological sequence entropy is strictly positive. Extending the proof to a more general setting, we further obtain that tame implies regular for almost automorphic group actions on compact spaces. In the converse direction, we show that even in the restrictive case of Euclidean cut and project schemes irregular model sets may be uniquely ergodic and have zero topological entropy. This provides ne…
