scholarly journals Topological entropy of sets of generic points for actions of amenable groups

2017 ◽  
Vol 61 (5) ◽  
pp. 869-880 ◽  
Author(s):  
Dongmei Zheng ◽  
Ercai Chen
Keyword(s):  
Topological Entropy ◽  
Amenable Groups ◽  
Generic Points

Tilings of amenable groups

2019 ◽  
Vol 2019 (747) ◽  
pp. 277-298 ◽  
Author(s):  
Tomasz Downarowicz ◽  
Dawid Huczek ◽  
Guohua Zhang
Keyword(s):  
Fixed Points ◽  
Topological Entropy ◽  
Finite Subset ◽  
Dimensional Space ◽  
Amenable Group ◽  
Free Action ◽  
Amenable Groups ◽  
Entropy Zero

Abstract We prove that for any infinite countable amenable group G, any {\varepsilon>0} and any finite subset {K\subset G} , there exists a tiling (partition of G into finite “tiles” using only finitely many “shapes”), where all the tiles are {(K,\varepsilon)} -invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of G (in the sense that the mappings, associated to elements of G other than the unit, have no fixed points) on a zero-dimensional space, such that the topological entropy of this action is zero.

scholarly journals Packing topological entropy for amenable group actions

2021 ◽  
pp. 1-35
Author(s):  
DOU DOU ◽  
DONGMEI ZHENG ◽  
XIAOMIN ZHOU
Keyword(s):  
Topological Entropy ◽  
Borel Probability Measure ◽  
Amenable Group ◽  
Borel Subset ◽  
Entropy Inequality ◽  
Probability Measures ◽  
Local Entropy ◽  
Compact Metric Space ◽  
Borel Probability ◽  
Generic Points

Abstract Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system $(X,G)$ , where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu $ coincides with the metric entropy if either $\mu $ is ergodic or the system satisfies a kind of specification property.

Transmission rate conditions for distributed filtering in sensor networks against eavesdropper

2021 ◽  
pp. 014233122110056
Author(s):  
Bingya Zhao ◽  
Ya Zhang
Keyword(s):  
Sensor Networks ◽  
Topological Entropy ◽  
Kalman Filtering ◽  
Transmission Rate ◽  
Necessary Condition ◽  
Estimation Errors ◽  
Positive Role ◽  
Filtering Algorithm ◽  
Wiretap Channels ◽  
Kalman Filtering Algorithm

This paper studies the distributed secure estimation problem of sensor networks (SNs) in the presence of eavesdroppers. In an SN, sensors communicate with each other through digital communication channels, and the eavesdropper overhears the messages transmitted by the sensors over fading wiretap channels. The increasing transmission rate plays a positive role in the detectability of the network while playing a negative role in the secrecy. Two types of SNs under two cooperative filtering algorithms are considered. For networks with collectively observable nodes and the Kalman filtering algorithm, by studying the topological entropy of sensing measurements, a sufficient condition of distributed detectability and secrecy, under which there exists a code–decode strategy such that the sensors’ estimation errors are bounded while the eavesdropper’s error grows unbounded, is given. For collectively observable SNs under the consensus Kalman filtering algorithm, by studying the topological entropy of the sensors’ covariance matrices, a necessary condition of distributed detectability and secrecy is provided. A simulation example is given to illustrate the results.

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

2021 ◽  
pp. 1-36
Author(s):  
ARIE LEVIT ◽  
ALEXANDER LUBOTZKY
Keyword(s):  
Ergodic Theorem ◽  
Abelian Groups ◽  
Wreath Products ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Amenable Groups ◽  
Pointwise Ergodic Theorem ◽  
Lamplighter Group ◽  
Invariant Random Subgroups

Abstract We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

Agricultural Multifunctionality and Care Farming: Insight from the UK

2015 ◽  
Vol 4 (1) ◽  
pp. 74-86
Author(s):  
Paul Custance ◽  
Keith Walley ◽  
Gaynor Tate ◽  
Goksel Armagan
Keyword(s):  
Learning Material ◽  
Multifunctional Agriculture ◽  
Economic Support ◽  
International Audience ◽  
The Uk ◽  
Generic Points ◽  
Care Farming ◽  
Compare And Contrast ◽  
Insight Into

The purpose of the article is to provide insight into care farming and the role that it may play in agriltural multifunctionality. The paper outlines three case studies of care farming in the UK to compare and contrast the roles that such organizations may play in multifunctional agriculture. Although the work has the obvious limitation of being based on case-study care farms that are based in the UK, the findings are sufficiently generic to serve as valuable learning material for those interested in the subject and located anywhere in the world. The main finding from this study is that care farming can take many different forms but still contribute to agricultural multifunctionality. The study also confirms the important roles that economic support and favourable legislation play in successful care farming. The paper concludes that care farming is a legitimate form of agricultural multifunctionality but reminds those interested in setting up or promoting care farms of the need for a supportive economic and legislative environment. The paper provides contemporary insight into the concept of care farming as a form of agricultural multifunctionality. A number of generic points are made that should be of value to an international audience of academics researching in this area as well as students studying care farming and agricultural multifunctionality, farmers considering diversifying into care farming and politicians working to create a political and economic environment that may support care farms.

scholarly journals Entropy on noncompact sets

2019 ◽  
Vol 7 (1) ◽  
pp. 29-37
Author(s):  
Jose S. Cánovas
Keyword(s):  
Topological Entropy ◽  
Topological Spaces ◽  
Continuous Maps

AbstractIn this paper we review and explore the notion of topological entropy for continuous maps defined on non compact topological spaces which need not be metrizable. We survey the different notions, analyze their relationship and study their properties. Some questions remain open along the paper.

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