Definition 1.11 ([53]). Let X be a sample space where X contains n events w1, w2, ..., wnwith probability p1, p2, ..., pn, respectively. The Shannon’s entropy with respect to X is defined by
H(p1, ..., pn) =− Xn
i=1
pilog(pi).
The following holds.
Theorem 1.12 ([53]). In the above definition, H 1
n, ..., 1 n
= log n
= max (
H(p1, ..., pn) : pi ≥ 0, Xn
i=1
pi = 1 )
.
Based on studies [53, 54], conducted in 1948, Shannon introduced a definition of entropy for use in information theory to establish the concept of “uncertain quantities” [83]. Later, in 1958, Kolmogorov used Shannon’s entropy to formalize the entropy on dynamical systems and opened the door to established the foundation of ergodic theory. This “Kolmogorov’s entropy” was improved and revised by his student Sinai in 1959. The revised version is called
“Kolmogorov-Sinai entropy” or “Measure-theoretic entropy”.
Definition 1.13 ([53, 75]). Suppose (X, Σ, µ) is a probability measure space, that is, X is a set, Σ is a σ-algebra of subsets of X, and µ is a probability on Σ. Let f : X → X be a measurable function. Arbitrarily choose a finite partition ¯A = {A1, ..., An}, that is, Ai ∈ Σ, Ai∩ Aj =∅, and Sn
i=1
Ai = X. The entropy of f with respect to ¯A is defined by
hµ f, ¯A
= lim sup
n→∞
H f−n( ¯A)|
n−1_
i=0
f−i( ¯A)
! ,
where ¯A∨ ¯B =
A∩ B : A ∈ ¯A, B ∈ ¯B
, and ¯A and ¯B are finite partitions. The Kolmogorov-Sinai entropy of f is defined by
hµ(f ) = sup
hµ(f, ¯A) : ¯A is the finite partition of X . Remark 1.14. If f is measure-preserving transformation, then
hµ(f, ¯A) = lim sup
for any finite partition ¯A of X. Although this equation is not more easily understood or interpreted than the foregoing definition, its equivalent condition can be more conveniently computed [53].
Modeled after Kolmogorov-Sinai entropy, “topological entropy” was defined by Adler, Konhein, and McAndrew for topologically conjugate invariance in 1965 [2].
Definition 1.15 ([2, 53, 54]). Suppose (X, T ) is a dynamical system, where X is a compact Hausdorff space and T : X → X is a continuous map. For an open cover ¯A of X, let N( ¯A) be defined as the smallest cardinality of a subcover of ¯A. If ¯A and ¯B are open covers of X, then their common refinement is defined as ¯A∨ ¯B ={A ∩ B : A ∈ ¯A, B ∈ ¯B}. Let
The topological entropy of (X, T ) is defined as the supremum htop(T ) = sup
htop(T, ¯A) : ¯A is the finite open cover of X .
Remark 1.16. N( ¯A) is finite because of compactness, and the limit exists for any open cover A.¯
If the space is compact metric, then the following definition is equivalent to the above notion [12] and it is more useful [4].
Definition 1.17 ([12, 15, 74]). Let f : X → X be a continuous map on the space X with metric d. A set S ⊂ X is called (n, ǫ)-separated for f for n a positive integer and ǫ > 0 provided that for every pair of distinct points x, y ∈ S, x 6= y, there is at least one k with 0 ≤ k < n such that d(fk(x), fk(y)) > ǫ. The number of different orbits of length n (as measured by ǫ) is defined by
r(n, ǫ, f ) = {#(S) : S ⊂ X is a (n, ǫ)-separated set for f }, where #(S) is the cardinality of elements in S. Let
htop(ǫ, f ) = lim sup
n→∞
log(r(n, ǫ, f ))
n ,
and define the topological entropy of f as htop(f ) = lim
ǫ→0,ǫ>0htop(ǫ, f ).
Remark 1.18. Here, only two definitions are introduced. See other references for topological entropy [9, 13, 29, 96].
Example 1.19 ([74]). Let f : S1 → S1 have a covering map F : R→ R given by F (x) = 2x.
This map is called the doubling map. The distance on S1 is the one inherited from R by taking x to x mod 1. Therefore, points near 1 are close to points near 0. Two points x and y stay within ǫ of each other for n−1 iterations of f if and only if |x−y| ≤ ǫ2−(n−1) (as points in R).
If points are separated by a distance of exactly ǫ2−(n−1) in [0, 1), then the maximum number of points is [ǫ−12n−1]. However, the last point to the right is close to the first point on the left when considered modulo 1 in S1, so there are [ǫ−12n−1]− 1 points in S1. These points can be spread apart slightly to make them (n, ǫ)-separated. Therefore, r (n, ǫ, f ) = [ǫ−12n−1]− 1, where [a] is the integer part of a. Then,
htop(ǫ, f ) = lim sup
n→∞
log ([ǫ−12n−1]− 1) n
= lim sup
n→∞
log (ǫ−1) + (n− 1) log(2) n
= log(2) for any ǫ > 0, so htop(f ) = log(2).
The following theorem relates the topological entropy to the Kolmogorov-Sinai entropy:
Theorem 1.20 ([24]). If K is compact and f : K → K is continuous, then htop(f ) = sup{hµ(f ) : µ is an ergodic measure with respect to f }.
Remark 1.21. An invariant measure µ satisfies the equation µ(f−t(K)) = µ(K), t > 0.
An invariant probability measure µ may be decomposable into several pieces, each of which is again invariant. If it is not so decomposable, then it is said to be ergodic.
Once the topological entropy has been computed, whether systems are chaotic or not is known.
Theorem 1.22 ([51]). If a continuous map of the interval has positive topological entropy, then it is chaotic according to the definition of Li and Yorke.
Remark 1.23 ([28, 51, 92]). The converse of this theorem is incorrect. Xiong [101] and Sm´ıtal [92] given counterexamples.
In general metric space, a similar result is obtained:
Theorem 1.24 ([10]). If the dynamical system (X, T ) has positive topological entropy, where X is compact metric space and T is surjective and continuous, then it is chaotic in the sense of Li and Yorke.
Consider the continuous map on the compact interval, the relationship between positive topological entropy and Devaney’s chaos is equivalent:
Theorem 1.25 ([48, 50, 51, 63]). Let f be a continuous map of a compact interval I to itself.
f has positive topological entropy if and only if f is chaotic in the sense of Devaney.
Let f : X → X and g : Y → Y be two maps. f and g are said to be topologically conjugate if there exists a homeomorphism h : X → Y such that h ◦ f = g ◦ h. The homeomorphism h is called a topological conjugacy. Mappings that are topologically conjugate have completely equivalent dynamics [21].
Adler, Konhein, and McAndrew established “topological entropy” for invariant of topo-logically conjugate, such that if two maps are topotopo-logically conjugate, then their topological entropies are equal as follows:
Theorem 1.26 ([2, 54, 74]). Let f : X → X and g : Y → Y be two continuous maps, where X and Y are invariant compact sets under f and g, respectively. That is, f (X) ⊂ X and g(Y )⊂ Y . If f and g are topologically conjugate, then htop(f ) = htop(g).
This basic result that is used to help calculate the entropy, relates the entropy of a map f to a power fk of f.
Theorem 1.27 ([74]). Assume f : X → X is uniformly continuous or X is compact, and k is an integer with k ≥ 1. Then htop(fk) = k· htop(f ).