scholarly journals Computing the Topological Entropy of Multimodal Maps via Min-Max Sequences

Entropy ◽  
2012 ◽  
Vol 14 (4) ◽  
pp. 742-768 ◽  
Author(s):  
José María Amigó ◽  
Rui Dilão ◽  
Ángel Giménez
Keyword(s):  
Topological Entropy

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.

scholarly journals A topological lens for a measure-preserving system

2010 ◽  
Vol 31 (1) ◽  
pp. 49-75 ◽  
Author(s):  
E. GLASNER ◽  
M. LEMAŃCZYK ◽  
B. WEISS
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Positive Entropy ◽  
Topological Properties ◽  
Topologically Transitive ◽  
Topological System ◽  
Zero Entropy ◽  
Weakly Mixing ◽  
Dynamical Properties

AbstractWe introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

Topological entropy of piecewise bimonotone skew products

2009 ◽  
Vol 15 (1) ◽  
pp. 53-69
Author(s):  
Franz Hofbauer ◽  
Peter Maličký ◽  
L'ubomír Snoha
Keyword(s):  
Topological Entropy ◽  
Skew Products

SIMPLE AND COMPLEX DYNAMICS FOR ONE-DIMENSIONAL MANIFOLD MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1351-1355
Author(s):  
VLADIMIR FEDORENKO
Keyword(s):  
Topological Entropy ◽  
Complex Dynamics ◽  
Periodic Points ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
Circle Maps ◽  
Interval Maps ◽  
Chain Recurrent Set ◽  
Recurrent Set

We give a characterization of complex and simple interval maps and circle maps (in the sense of positive or zero topological entropy respectively), formulated in terms of the description of the dynamics of the map on its chain recurrent set. We also describe the behavior of complex maps on their periodic points.

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