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Relative Entropy, Asymptotic Pairs and Chaos
Author(s) -
Zhang Guohua
Publication year - 2006
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610705022520
Subject(s) - mathematics , topological entropy , infimum and supremum , topological entropy in physics , entropy (arrow of time) , conditional entropy , statistical physics , joint quantum entropy , quantum relative entropy , conditional quantum entropy , discrete mathematics , combinatorics , topology (electrical circuits) , pure mathematics , principle of maximum entropy , maximum entropy thermodynamics , physics , statistics , quantum entanglement , quantum discord , thermodynamics , quantum , quantum mechanics , topological quantum number
In this paper, we prove that positive conditional entropy implies the existence of asymptotic pairs and scrambled sets on fibers. Moreover, we introduce the notion of conditional topological entropy for a subset using Bowen's definition of separated and spanning sets, and prove that the conditional topological entropy of a system relative to a factor is the supremum of conditional topological entropy of its scrambled sets on fibers.