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The topological entanglement entropy [1] [2] [3] or topological entropy, usually denoted by , is a number characterizing many-body states that possess topological order. A non-zero topological entanglement entropy reflects the presence of long range quantum entanglements in a many-body quantum state.
In mathematics, the topological entropy of a topological dynamical system is a nonnegative extended real number that is a measure of the complexity of the system. Topological entropy was first introduced in 1965 by Adler , Konheim and McAndrew.
- History
- Definitions
- Interpretation
- Basic Properties of Topological Entropy
- Relation with Kolmogorov-Sinai Entropy
- Topological Entropy in Some Special Cases
- Related Notions
- Generalizations of Topological Entropy
- References
The original definition was introduced by Adler, Konheim and McAndrew in 1965 [AKM]. Their idea to assign a number to an open cover to measure its size was inspired by Kolmogorov and Tihomirov [KT] (1961). Then to define topological entropy for continuous maps they strictly imitated the definition of Kolmogorov-Sinai entropy of a measure preserving...
By Adler, Konheim and McAndrew
For an open cover \mathcal{U}U of XX (i.e., a family of open sets whose union is XX), let N(\mathcal{U})N(U) denote the smallest cardinality of a subcover of \mathcal{U}U (i.e., a subfamily of \mathcal{U}U whose union still equals XX). By compactness, N(\mathcal{U})N(U) is always finite. If \mathcal{U}U and \mathcal{V}V are open covers of XX then \mathcal{U}\vee\mathcal{V} = \{U\cap V:U\in\mathcal{U}, V\in\mathcal{V}\}U∨V={U∩V:U∈U,V∈V} is called their common refinement. Let \mathcal{U}^n=\mat...
By Bowen and Dinaburg
Assume for simplicity that XX is metric (otherwise a uniform structure can be used). A set E \subset XE⊂X is said to be (n,\varepsilon)(n,ε)-separated, if for every x, y\in Ex,y∈E with x\neq yx≠y there is i\in\{0,1,\dots,n-1\}i∈{0,1,…,n−1} such that d(Tix,Tiy)\ge\varepsilon\ .d(Tix,Tiy)≥ε. Let s(n,\varepsilon)s(n,ε) be the maximal cardinality of an (n,\varepsilon)(n,ε)-separated set in X\ .X. Again, by compactness, this number is always finite. One defines\overline h(\varepsilon,T)=\limsup_...
Equality between the two notions
It is not hard to see that if \mathcal{U} is an open cover with all elements of diameter at most \varepsilon and Lebesgue number 2\delta thens(n,\varepsilon) \le N(\mathcal{U}^n) \le s(n,\delta),which not only implies that h_B(T) = h_A(T) but also that the same number h_B(T) is obtained if \overline h is replaced by \underline h defined using \liminf in place of \limsup\ . From now on we will use h(T) to denote either h_A(T) or h_B(T)\ .
The interpretation of the number s(n,\varepsilon) is the following: suppose one observes the system with a device of resolution \varepsilon\ , i.e., two points are distinguished only if the distance between them is at least \varepsilon\ . Then, after n steps of the evolution of the system, the observer will be able to distinguish at most s(n,\varep...
h(T)\ge h(S) if (Y,S) is a topological factor of (X,T)\ , i.e., \phi\circ T=S\circ \phi, where \phi:X\to Yis a continuous surjection;
The relation between topological entropy and measure theoretic entropy is established by the Variational Principle, which asserts thath(T)=\sup\{h_\mu(T):\mu\in \mathcal{P}_T(X)\},i.e., topological entropy equals the supremum of the Kolmogorov-Sinai entropies h_\mu(T)\ , where \mu ranges over all T-invariant Borel probability measures on X\ . In ma...
In some special cases of topological dynamical systems, topological entropy can be computed in a more direct way. 1. If (X,T) is a subshift (also called a symbolic system), i.e., T is the shift map on a closed shift invariant collection X of (one-sided or two-sided) sequences of symbols belonging to a finite alphabet, then h(T) = \lim_{n\to\infty}\...
Topological tail entropy and symbolic extension entropy
1. In 1976, Misiurewicz [M] introduced an entropy-related parameter h^*(T)\ , which he called the topological conditional entropy. It equals the infimum over all finite covers \mathcal U of the conditional entropy given \mathcal U (the definition of this conditional entropy is similar to Bowen's definition of topological entropy, only one counts (n,\epsilon)-separated points inside an element of the cover U^n). Because the infimum does not depend on any conditioning parameter (cover) any more...
Mean topological dimension
Lindenstraus and Weiss [L], [LW] introduced a new invariant, proposed by Gromov, called mean topological dimension mdim(X,T)\ , which solved a long-standing imbedding problem of topological dynamics. For a number of dynamical systems this invariant can be calculated. If the topological dimension of X is finite, or if (X,T) has finite topological entropy then mdim(X,T)= 0\ , so the new invariant is good for distinguishing between systems where the topological dimension of Xis infinite or (X,T)...
Topological entropy and chaos
Topological dynamical systems of positive entropy are often considered topologically chaotic. Positive entropy always implies Li-Yorke chaos defined as the existence of an uncountable scrambled set (see Blanchard et al. [BGKM]). For interval maps the condition h(T)>0 is equivalent to distributional chaos as defined by Schweizer and Smital in 1994 [SS]. Nonetheless, some systems of zero topological entropyreveal very complicated behavior, in particular, they may be Li-Yorke or distributionally...
Complexity
The parameter which controls the subexponential growth of the number of distinguishable orbits in the zero entropy case is topological complexity(see Blanchard et al. [BHM]).
Pressure
A notion more general than topological entropy, depending also on a function f:X\to\Bbb R\ , is pressure. Topological entropy is the pressure for f\equiv 0\ .
Topological entropy for flows
For a flow \varphi:\Bbb R\times X\to X the topological entropy is defined as the entropy of the time-one maph(\varphi)=h(\varphi^1)\ . Since for flows a notion that is used more often than conjugacy is equivalence, which admits rescaling of time, topological entropy basically distinguishes only between flows with zero, positive finite, and infinite entropy.
[AKM] R. L. Adler, A. G. Konheim and M. H. McAndrew: Topological entropy, Trans. Amer. Math. Soc. 114 (1965) 309-319 [AM] R. L. Adler, and B. Marcus: Topological entropy and the equivalence of dynamical systems, Mem. Amer. Math. Soc. 219 (1979) [ACH] R. L. Adler, D. Coppersmith, and M. Hassner: Algorithmsfor sliding block: An application of symboli...
Topological entropy measures the evolution of distinguishable orbits over time, thereby providing an idea of how complex the orbit structure of a system is. Entropy distinguishes a dynamical system where points that are close together remain close from a dynamical system in which groups of points move farther. In the rst section, we develop the ...
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Topological entropy. x8.1 Introduction. In the previous lecture we de ned the entropy of a measure-preserving trans-formations on a probability space. In this lecture we study entropy in the context of continuous transformations of compact metric spaces. In partic-ular, given a continuous transformation T : X !
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May 2, 2022 · In this chapter, we introduced the topological entropy of a map. This can be defined for very general maps and spaces, but here we specialize to pseudo-Anosov maps on punctured disks: The entropy is an upper bound for the rate of growth of words in π 1 (S), under repeated action of the induced map ϕ ∗.
The topological entanglement entropy, usually denoted by γ, is a number characterizing many-particle states that possess topological order. The short form topological entropy is often used, although the same name in ergodic theory refers to an unrelated mathematical concept (see topological entropy).
