Entropy for actions of free groups under bounded orbit-equivalence

A note on entropy of auto-equivalences: lower bound and the case of orbifold projective lines

Entropy of categorical dynamics is defined by Dmitrov-Haiden-Katzarkov-Kontsevich. Motivated by the fundamental theorem of the topological entropy due to Gromov-Yomdin, it is natural to ask an equality between the entropy and the spectral radius of induced morphisms on the numerical Grothendieck group. In this paper, we add two results on this equality: the lower bound in a general setting and the equality for orbifold projective lines.

Number of bounded distance equivalence classes in hulls of repetitive Delone sets

Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that the hull of a repetitive Delone set with finite local complexity has either one equivalence class or uncountably many. A very similar result is proven in arXiv:2011.00106 [math.MG].

Entropy, Shannon orbit equivalence, and sparse connectivity

We say that two free probability-measure-preserving actions of countable groups are Shannon orbit equivalent if there is an orbit equivalence between them whose associated cocycle partitions have finite Shannon entropy. We show that if the acting groups are sofic and each has a w-normal amenable subgroup which is neither locally finite nor virtually cyclic then Shannon orbit equivalence implies that the actions have the same maximum sofic entropy. This extends a result of Austin beyond the finitely generated amenable setting and has the consequence that two Bernoulli actions of a group with th…

Weak equivalence of stationary actions and the entropy realization problem

We introduce the notion of weak containment for stationary actions of a countable group and define a natural topology on the space of weak equivalence classes. We prove that Furstenberg entropy is an invariant of weak equivalence, and moreover that it descends to a continuous function on the space of weak equivalence classes.

Behaviour of entropy under bounded and integrable orbit equivalence

Let $G$ and $H$ be infinite finitely generated amenable groups. This paper studies two notions of equivalence between actions of such groups on standard Borel probability spaces. They are defined as stable orbit equivalences in which the associated cocycles satisfy certain tail bounds. In `integrable stable orbit equivalence', the length in $H$ of the cocycle-image of an element of $G$ must have finite integral over its domain (a subset of the $G$-system), and similarly for the reverse cocycle. In `bounded stable orbit equivalence', these functions must be essentially bounded in terms of the l…

Ultraproducts, weak equivalence and sofic entropy

In this work, we study pmp actions of countable groups on arbitrary diffuse probability spaces under the point of view of weak equivalence. We will show that any such an action is weakly equivalent to an action on a standard probability space. We also propose a metric on the space of actions modulo weak equivalence which is equivalent to the topology of Abért and Elek. We will give a simpler proof of the compactness of the space, showing that convergence is characterized by ultraproducts. Using this topology, we will show that a profinite action is weakly equivalent to an ultraproduct of finit…

Entropy, products, and bounded orbit equivalence

We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, then the actions have the same sofic topological entropy. This fact is then used to show that if two free uniquely ergodic and entropy regular probability-measure-preserving actions of such groups are boundedly orbit equivalent then the actions have the same s…

Superrigidity, measure equivalence, and weak Pinsker entropy

We show that the class $\mathscr{B}$, of discrete groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli actions, is invariant under measure equivalence. We generalize this to the setting of discrete p.m.p. groupoids, and as a consequence we deduce that any nonamenable lattice in a product of two noncompact, locally compact second countable groups, must belong to $\mathscr{B}$. We also introduce a measure-conjugacy invariant called Weak Pinsker entropy and show that, if G is a group in the class $\mathscr{B}$, then Weak Pinsker entropy is an orbit-equivalence…

用于定量轨道等价的Odomutants与灵活性结果

Odomutants and flexibility results for quantitative orbit equivalence

本文提出了通过扭曲odometer轨道构建的odomutants系统，用于研究定量轨道等价的灵活性，并证明了多个关于熵保持和轨道等价关系的最优结果。

Variational principles for t-entropy, the spectral potential of transfer operator, and entropy statistic theorem are equivalent

For any transfer operator we establish the equivalence of variational principles for $t$-entropy, the spectral potential and entropy statistic theorem and give new proofs for all these statements.

通过增长率比较研究$\mu$-二分性、有界增长与谱性质的相互作用

The interplay of $μ$-dichotomy, bounded growth, and spectral properties via growth rate comparisons

该研究针对非自治线性系统，通过引入增长率比较准则与等价关系，阐明μ-二分性与μ-有界增长的相互作用，揭示谱随速率的伸缩规律，证明等价意义下唯一满足双性质的速率存在性，并实现二分谱分类，结果适用于离散与连续时间场景。

Delone集的双Lipschitz与有界位移等价性

BiLipschitz and bounded displacement equivalence of Delone sets

本文综述Delone集的双Lipschitz与有界位移等价性，重点研究非周期序相关实例，涉及经典结果、相关判据及替换铺砌、截集等类别的扩展情况。

Continuously many bounded displacement non-equivalences in substitution tiling spaces

We consider substitution tilings in R^d that give rise to point sets that are not bounded displacement (BD) equivalent to a lattice and study the cardinality of BD(X), the set of distinct BD class representatives in the corresponding tiling space X. We prove a sufficient condition under which the tiling space contains continuously many distinct BD classes and present such an example in the plane. In particular, we show here for the first time that this cardinality can be greater than one.

A dichotomy for bounded displacement equivalence of Delone sets

We prove that in every compact space of Delone sets in $\mathbb{R}^d$ which is minimal with respect to the action by translations, either all Delone sets are uniformly spread, or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty--Fell topology, which is the natural topology on the space of closed subsets of $\mathbb{R}^d$. This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all min…

切割-投影集的有界距离等价性与等分解性

Bounded distance equivalence of cut-and-project sets and equidecomposability

本文研究了切割-投影集的有界距离等价性与等分解性关系，揭示了等分解性在确定有界距离等价类中的关键作用。

有限划分的预紧性、零最大模式熵和有界平均复杂度等价性

The equivalence of precompactness, zero maximal pattern entropy and bounded mean complexity for finite partitions

本文研究了有限划分的低复杂度类型，证明了预紧性、零最大模式熵、有界平均复杂度和均等连续性之间的等价关系。

Zero entropy and bounded topology

We study the existence of Riemannian metrics with zero topological entropy on a closed manifold M with infinite fundamental group. We show that such a metric does not exist if there is a finite simply connected CW complex which maps to M in such a way that the rank of the map induced in the pointed loop space homology grows exponentially. This result allows us to prove in dimensions four and five, that if M admits a metric with zero entropy then its universal covering has the rational homotopy type of a finite elliptic CW complex. We conjecture that this is the case in every dimension.

Equivalence classes of codimension one cut-and-project nets

We prove that in any totally irrational cut-and-project setup with codimension (internal space dimension) one, it is possible to choose sections (windows) in non-trivial ways so that the resulting sets are bounded displacement to lattices. Our proof demonstrates that for any irrational $α$, regardless of Diophantine type, there is a collection of intervals in $\mathbb{R}/\mathbb{Z}$ which is closed under translation, contains intervals of arbitrarily small length, and along which the discrepancy of the sequence $\{nα\}$ is bounded above uniformly by a constant.

Cohomology groups invariant under continuous orbit equivalence

By the work of Brodzki-Niblo-Nowak-Wright and Monod, topological amenability of a continuous group action can be characterized using uniformly finite homology groups or bounded cohomology groups associated to this action. We show that (certain variations of) these groups are invariants for topologically free actions under continuous orbit equivalence.

Minimal entropy and collapsing with curvature bounded from below

We show that if a closed manifold M admits an F-structure (possibly of rank 0) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial circle action. As a corollary we obtain that the simplicial volume of a colsed manifold admitting an F-structure is zero. We also show that if M admits an F-structure then it collapses with curvature bounded from below. This is turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative. We show that F-structures of rank zero appear rather frequently:every compact comple…

Continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz--Krieger algebras

We study KMS states for gauge actions with potential functions on Cuntz--Krieger algebras whose underlying one-sided topological Markov shifts are continuous orbit equivalent. As a result, we have a certain relationship between topological entropy of continuous orbit equivalent one-sided topological Markov shifts.

Further equivalences and semiglobal versions of integral input to state stability

This paper continues the study of the integral input-to-state stability (IISS) property. It is shown that the IISS property is equivalent to one which arises from the consideration of mixed norms on states and inputs, as well as to the superposition of a ``bounded energy bounded state'' requirement and the global asymptotic stability of the unforced system. A semiglobal version of IISS is shown to imply the global version, though a counterexample shows that the analogous fact fails for input to state stability (ISS). The results in this note complete the basic theoretical picture regarding IIS…

非交换 Poisson 边界、超积与熵

Noncommutative Poisson boundaries, ultraproducts and entropy

本文利用 von Neumann 代数超积构造带迹情形的非交换 Poisson 边界，并据此完善 Kaimanovich-Vershik 非交换熵基本定理，证明两类边界等价关系。

交换幺半群的有界到一作用的超有限性

Hyperfiniteness of bounded-to-one actions of commutative monoids

该研究针对可数个两两交换的Borel函数生成等价关系的开放问题，在函数为有界到一的情形下给出肯定解答，推广了Gao-Jackson的相关定理。

Thermodynamic Formalism for Haar systems in Noncommutative Integration: transverse functions and entropy of transverse measures

We consider here a class of groupoids obtained via an equivalence relation (the subgroupoids of pair groupoids). We generalize to Haar Systems in these groupoids some results related to entropy and pressure which are well known in Thermodynamic Formalism. We introduce a transfer operator, where the equivalence relation (which defines the groupoid) plays the role of the dynamics and the corresponding transverse function plays the role of the {\it a priori} probability. We also introduce the concept of invariant transverse probability and of entropy for an invariant transverse probability, as we…