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- 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. Their definition was modelled after the definition of the Kolmogorov–Sinai, or metric entropy.
en.wikipedia.org/wiki/Topological_entropy
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.
Nov 21, 2019 · 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. Share.
Oct 15, 2017 · The Kolmogorov–Sinai entropy is a metric, measure-theoretic entropy (rate) based on the Shannon entropy, while the topological entropy is a combinatorial entropy that gives the growth rate of the number of allowed sequences.
- 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 U of X (i.e., a family of open sets whose union is X), let N(U) denote the smallest cardinality of a subcover of U (i.e., a subfamily of U whose union still equals X). By compactness, N(U) is always finite. If U and V are open covers of X then U∨V={U∩V:U∈U,V∈V} is called their common refinement. Let Un=U∨T−1U∨⋯∨T−n+1U, whereT−kU={T−kU:U∈U}. Using a subadditivity argument, one shows that the limith(U,T)=limn→∞logN(Un)nexists for any open cover U (and equals infn∈NlogN(Un)n)....
By Bowen and Dinaburg
Assume for simplicity that X is metric (otherwise a uniform structure can be used). A set E⊂X is said to be (n,ε)-separated, if for every x,y∈E with x≠y there is i∈{0,1,…,n−1} such that d(Tix,Tiy)≥ε. Let s(n,ε) be the maximal cardinality of an (n,ε)-separated set in X. Again, by compactness, this number is always finite. One definesh¯¯(ε,T)=lim supn→∞logs(n,ε)n.The topological entropy is obtained ashB(T)=supε>0h¯¯(ε,T)=limε→0h¯¯(ε,T). It should be noted that s(n,ε) can be substituted in Bowen...
Equality between the two notions
It is not hard to see that if U is an open cover with all elements of diameter at most ε and Lebesgue number 2δ thens(n,ε)≤N(Un)≤s(n,δ),which not only implies that hB(T)=hA(T) but also that the same number hB(T) is obtained if h¯¯ is replaced by h− defined using lim inf in place of lim sup. From now on we will use h(T) to denote either hA(T) or hB(T).
The interpretation of the number s(n,ε) is the following: suppose one observes the system with a device of resolution ε, i.e., two points are distinguished only if the distance between them is at least ε. Then, after n steps of the evolution of the system, the observer will be able to distinguish at most s(n,ε) different orbits. Thus, the value h¯¯...
h(T)≥h(S) if (Y,S) is a topological factor of (X,T), i.e., ϕ∘T=S∘ϕ, where ϕ:X→Yis a continuous surjection;
The relation between topological entropy and measure theoretic entropy is established by the Variational Principle, which asserts thath(T)=sup{hμ(T):μ∈PT(X)},i.e., topological entropy equals the supremum of the Kolmogorov-Sinai entropies hμ(T), where μ ranges over all T-invariant Borel probability measures on X. In many cases (for instance in asymp...
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)=limn→∞log#Bnn, where...
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 U of the conditional entropy given U (the definition of this conditional entropy is similar to Bowen's definition of topological entropy, only one counts (n,ϵ)-separated points inside an element of the cover Un). Because the infimum does not depend on any conditioning parameter (cover) any more, h∗(T) is recently called th...
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)has i...
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→R, is pressure. Topological entropy is the pressure for f≡0.
Topological entropy for flows
For a flow φ:R×X→X the topological entropy is defined as the entropy of the time-one maph(φ)=h(φ1). 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...
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.
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 !
The topological entropy is defined as the limit (asε→0) of the exponential growth-rate of s n(ε): h top(T) = lim ε→0 limsup n→∞ 1 n logs n(ε). (3) Note that s n(ε 1) ⩾ s n(ε 2) if ε 1 ⩽ ε 2, so limsup n 1 n logs n(ε) is a decreasing function in ε, and the limit as ε→0 indeed exists.
