A
topological dynamical system consists of a
Hausdorff topological space X (usually assumed to be
compact) and a
continuous self-map
f :
X →
X. Its
topological entropy is a nonnegative
extended real number that can be defined in various ways, which are known to be equivalent.
Definition of Adler, Konheim, and McAndrew Let
X be a compact Hausdorff topological space. For any finite open
cover C of
X, let
H(
C) be the
logarithm (usually to base 2) of the smallest number of elements of
C that cover
X. For two covers
C and
D, let C \vee D be their (minimal) common refinement, which consists of all the non-empty intersections of a set from
C with a set from
D, and similarly for multiple covers. For any
continuous map f:
X →
X, the following limit exists: : H(f,C) = \lim_{n\to\infty} \frac{1}{n} H(C\vee f^{-1}C\vee \ldots\vee f^{-n+1}C). Then the
topological entropy of
f, denoted
h(
f), is defined to be the
supremum of
H(
f,
C) over all possible finite covers
C of
X.
Interpretation The parts of
C may be viewed as symbols that (partially) describe the position of a point
x in
X: all points
x ∈
Ci are assigned the symbol
Ci . Imagine that the position of
x is (imperfectly) measured by a certain device and that each part of
C corresponds to one possible outcome of the measurement. H(C\vee f^{-1}C\vee \ldots\vee f^{-n+1}C) then represents the logarithm of the minimal number of "words" of length
n needed to encode the points of
X according to the behavior of their first
n − 1 iterates under
f, or, put differently, the total number of "scenarios" of the behavior of these iterates, as "seen" by the partition
C. Thus the topological entropy is the average (per iteration) amount of
information needed to describe long iterations of the map
f.
Definition of Bowen and Dinaburg This definition uses a
metric on
X (actually, a
uniform structure would suffice). This is a narrower definition than that of Adler, Konheim, and McAndrew, as it requires the additional metric structure on the topological space (but is independent of the choice of metrics generating the given topology). However, in practice, the Bowen-Dinaburg topological entropy is usually much easier to calculate. Let (
X,
d) be a
compact metric space and
f:
X →
X be a
continuous map. For each
natural number n, a new metric
dn is defined on
X by the formula :d_n(x,y)=\max\{d(f^i(x),f^i(y)): 0\leq i Given any
ε > 0 and
n ≥ 1, two points of
X are
ε-close with respect to this metric if their first
n iterates are
ε-close. This metric allows one to distinguish in a neighborhood of an orbit the points that move away from each other during the iteration from the points that travel together. A subset
E of
X is said to be '
(n
, ε
)-separated' if each pair of distinct points of
E is at least
ε apart in the metric
dn. Denote by
N(
n,
ε) the maximum
cardinality of an (
n,
ε)-separated set. The
topological entropy of the map
f is defined by :h(f)=\lim_{\epsilon\to 0} \left(\limsup_{n\to \infty} \frac{1}{n}\log N(n,\epsilon)\right).
Interpretation Since
X is compact,
N(
n,
ε) is finite and represents the number of distinguishable orbit segments of length
n, assuming that we cannot distinguish points within
ε of one another. A straightforward argument shows that the limit defining
h(
f) always exists in the
extended real line (but could be infinite). This limit may be interpreted as the measure of the average exponential growth of the number of distinguishable orbit segments. In this sense, it measures complexity of the topological dynamical system (
X,
f). Rufus Bowen extended this definition of topological entropy in a way which permits
X to be non-compact under the assumption that the map
f is
uniformly continuous. == Properties ==