Peng Sun
We show that for every topological dynamical system with the approximate product property, zero topological entropy is equivalent to unique ergodicity. Equivalence of minimality is also proved under a slightly stronger condition. Moreover, we show that unique ergodicity implies the approximate product property if the system has periodic points.
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Entropy and strict ergodicity for multidimensional Toeplitz subshifts
本研究通过自洽构造，证明了具有任意规定熵的严格遍历Toeplitz Z^d-子移位存在，且其极大等连续因子均为Z^d上的通用里程计。
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.
We show that a class of robustly transitive diffeomorphisms originally described by Mañé are intrinsically ergodic. More precisely we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic, but nevertheless have constant entropy and isomorphic unique measures of maximal entropy.
Any coded subshift X defined by a set C of code words contains a subshift, which we call L, consisting of limits of single code words. We show that when C satisfies a unique decomposition property, the topological entropy h(X) of X is determined completely by h(L) and the number of code words of each length. More specifically, we show that h(X) = h(L) exactly when a certain infinite series is less than or equal to 1, and when that series is greater than 1, we give a formula for h(X_C). In the latter case, an immediate corollary (using results of Climenhaga and Thompson) is that X has a unique …
A classification of intrinsic ergodicity for recognisable random substitution systems
本文研究了由随机替换生成的动力系统，探讨了内在ergodic性和多最大熵测度的分类，证明了最大熵测度通过适当的对称关系不变性分类，并给出了内在ergodic性的充分条件。
For strongly positively recurrent countable state Markov shifts, we bound the distance between an invariant measure and the measure of maximal entropy in terms of the difference of their entropies. This extends an earlier result for subshifts of finite type, due to Kadyrov. We provide a similar bound for equilibrium measures of strongly positively recurrent potentials, in terms of the pressure difference. For measures with nearly maximal entropy, we have new, and sharp, bounds. The strong positive recurrence condition is necessary.
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…
We give a short proof of the unique ergodicity of the strong stable foliation of the geodesic flow on the frame bundle of a hyperbolic manifold admitting a finite measure of maximal entropy. Equivalently, let G = S0o(n, 1), $Γ$ \textless{} G be a discrete subgroup of G, and G = N AK the Iwasawa decomposition of G. If the geodesic flow on $Γ$\G admits a finite measure of maximal entropy, we prove that the action of N on $Γ$\G by right multiplication admits a unique invariant measure supported on points whose A-orbit does not diverge.
Let $ β$ be a real number less than -1. In this paper, we prove the uniqueness of the measure with maximal entropy of the negative $β$-shift. Endowed with the shift, this symbolic dynamical system is coded under certain conditions, but in all cases, it is shown that the measure with maximal entropy is carried by a support coded by a recurrent positive code. One of the difference between the positive and the negative $β$-shift is the existence of gaps in the system for certain negative values of $ β$ . These are intervals of negative $β$-representations (cylinders) negligible with respect to th…
We study joint quasimodes of the Laplacian and one Hecke operator on compact congruence surfaces, and give conditions on the orders of the quasimodes that guarantee positive entropy on almost every ergodic component of the corresponding semiclassical measures. Together with the measure classification result of the second-named author, this implies Quantum Unique Ergodicity for such functions. Our result is optimal with respect to the dimension of the space from which the quasi-mode is constructed. We also study equidistribution for sequences of joint quasimodes of the two partial Laplacians on…
We prove a $C^1$ version of a conjecture by Pugh and Shub: among partially hyperbolic volume-preserving $C^r$ diffeomorphisms, $r>1$, the stably ergodic ones are $C^1$-dense. To establish these results, we develop new perturbation tools for the $C^1$ topology: linearization of horseshoes while preserving entropy, and creation of "superblenders" from hyperbolic sets with large entropy.
We apply the techniques of our previous paper to study joint eigenfunctions of the Laplacian and one Hecke operator on compact congruence surfaces, and joint eigenfunctions of the two partial Laplacians on compact quotients of $\mathbb{H}\times\mathbb{H}$. In both cases, we show that quantum limit measures of such sequences of eigenfunctions carry positive entropy on almost every ergodic component. Together with prior work of the second named author, this implies Quantum Unique Ergodicity for such functions.
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…
We show that every subshift factor of a ($-β$)-shift is intrinsically ergodic, when $β\geq \frac{1+\sqrt{5}}{2}$ and the ($-β$)-expansion of $1$ is not periodic with odd period. Moreover, the unique measure of maximal entropy satisfies a certain Gibbs property. This is an application of the technique established by Climenhaga and Thompson to prove intrinsic ergodicity beyond specification. We also prove that there exists a subshift factor of a ($-β$)-shift which is not intrinsically ergodic in the cases other than the above.
Shift spaces with the specification property are intrinsically ergodic, i.e. they have a unique measure of maximal entropy. This can fail for shifts with the weaker almost specification property. We define a new property called one-sided almost specification, which lies in between specification and almost specification, and prove that it guarantees intrinsic ergodicity if the corresponding mistake function g is bounded. We also show that uniqueness may fail for unbounded g such as log log n. Our results have consequences for almost specification: we prove that almost specification with g=1 imp…
Using the idea of local entropy theory, we characterize the sequence entropy tuple via mean forms of the sensitive tuple in both topological and measure-theoretical senses. For the measure-theoretical sense, we show that for an ergodic measure-preserving system, the $μ$-sequence entropy tuple, the $μ$-mean sensitive tuple and the $μ$-sensitive in the mean tuple coincide, and give an example to show that the ergodicity condition is necessary. For the topological sense, we show that for a certain class of minimal systems, the mean sensitive tuple is the sequence entropy tuple.
We consider the geodesic flow of reversible Finsler metrics on the 2-sphere and the 2-torus, whose geodesic flow has vanishing topological entropy. Following a construction of A. Katok, we discuss examples of Finsler metrics on both surfaces, which have large ergodic components for the geodesic flow in the unit tangent bundle. On the other hand, using results of J. Franks and M. Handel, we prove that ergodicity and dense orbits cannot occur in the full unit tangent bundle of the 2-sphere, if the Finsler metric has positive flag curvatures and at least two closed geodesics. In the case of the 2…
Intrinsic ergodicity for $\mathfrak{B}$-free integers in number fields
研究数域中无\(\mathfrak{B}\)整数的内在遍历性，证明相关子转移是内在遍历的，给出唯一最大熵测度。这是一维以上此类系统首个证明，依赖他人工作，还解决了相关情况，且给出基础刚性陈述的两个独立证明。
We construct a family of shift spaces with almost specification and multiple measures of maximal entropy. This answers a question from Climenhaga and Thompson [Israel J. Math. 192 (2012), no. 2, 785--817]. Elaborating on our examples we also prove that some sufficient conditions for every subshift factor of a shift space to be intrinsically ergodic given by Climenhaga and Thompson are in some sense best possible, moreover, the weak specification property neither implies intrinsic ergodicity, nor follows from almost specification. We also construct a dynamical system with the weak specification…
We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of $GL(2,\mathbb{Q}_p)$ for automorphic forms belonging to the principal series. We interpret this conclusion in terms of the equidistribution of eigenfunctions on covers of a fixed regular graph or along nested sequences of regular graphs. Our results are the first of their kind on any p-adic arithmetic quotient. They may be understood as analogues of Lindenstrauss's theorem on the equidistribution of Maass forms on a compact arithmetic surface. The new ingredients here include the introduction of a repre…
Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random perturbations on the dynamics of CA. As models of computation, they can be used to study the reliability of computation against noise. We consider various families of CA (nilpotent, permutive, gliders, CA with a spreading symbol, surjective, algebraic) and prove that they are highly unstable against noise, meaning that they forget their initial conditions under slightes…
Dynamical Systems on Compact Metrizable Groups
本文将 Berg 关于最大熵定理和测度卷积遍历性的结论推广至满射同态情形，证明了紧致可度量群上 Haar 扩张测度熵的不等式关系，并论证了不相交遍历系统的测度卷积在满射同态下保持遍历性。
Ergodic measures of intermediate entropies for amenable group actions with an approximate product property
本文针对无限可数顺从群连续作用，引入近似乘积性质，证明其蕴含不变测度的熵稠密性与几乎熵可逼近性，并在渐近熵扩张条件下提升为熵可逼近性，进而实现中间熵遍历测度的完全实现。
Isomorphisms and slow entropy of deterministic $[T,T^{-1}]$ systems
本文研究确定性$[T,T^{-1}]$系统的同构刚性，证明斜积同构条件，并构造任意低慢熵及违反慢熵变分原理的新例子。
On a Family of Random Noble Means Substitutions
本文推广了本原替换规则局部混合的概念，研究由有限个定义同一双侧离散动力壳的本原替换构成的noble means族，给出其随机情形下拓扑熵、遍历性及衍射测度谱型的相关结果。
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