Topological entropy bounds measure-theoretic entropy

1971 ◽  
pp. 69-84 ◽  
Author(s):  
L. Wayne Goodwyn
Keyword(s):  
Topological Entropy ◽  
Entropy Bounds

scholarly journals Finite entropy for multidimensional cellular automata

2008 ◽  
Vol 28 (4) ◽  
pp. 1243-1260 ◽  
Author(s):  
TOM MEYEROVITCH
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Topological Entropy ◽  
Polynomial Growth ◽  
Countable Group ◽  
Entropy Bounds ◽  
Finite Set ◽  
Multidimensional Cellular Automata

AbstractLet $X=S^{\mathbb {G}}$ where $\mathbb {G}$ is a countable group and S is a finite set. A cellular automaton (CA) is an endomorphism T:X→X (continuous, commuting with the action of $\mathbb {G}$). Shereshevsky [Expansiveness, entropy and polynomial growth for groups acting on subshifts by automorphisms. Indag. Math. (N.S.)4(2) (1993), 203–210] proved that for $\mathbb {G}=\mathbb {Z}^d$ with d>1 no CA can be forward expansive, raising the following conjecture: for $G=\mathbb {Z}^d$, d>1, the topological entropy of any CA is either zero or infinite. Morris and Ward [Entropy bounds for endomorphisms commuting with K actions. Israel J. Math. 106 (1998), 1–11] proved this for linear CAs, leaving the original conjecture open. We show that this conjecture is false, proving that for any d there exists a d-dimensional CA with finite, non-zero topological entropy. We also discuss a measure-theoretic counterpart of this question for measure-preserving CAs.

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.

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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