Topological entropy

A concept in topological dynamics and ergodic theory analogous to the metric entropy of dynamical systems (introduced in [1]). For an open covering $ \mathfrak A $ of a compactum $ X $, let $ H ( \mathfrak A ) $ denote the logarithm (usually to base 2) of the smallest number of elements of the covering that can cover $ X $. If $ S: X \rightarrow X $ is a continuous mapping, then the limit

$$ h ( S, \mathfrak A ) = \ \lim\limits _ {n \rightarrow \infty } \ { \frac{1}{n} } H ( \mathfrak A \lor S ^ {-1} \mathfrak A \lor \dots \lor S ^ {- n + 1 } \mathfrak A ) $$

exists, where $ \mathfrak A \lor \mathfrak B $ is the covering whose elements are all the non-empty intersections of the elements of $ \mathfrak A $ and $ \mathfrak B $. The topological entropy $ h _ { \mathop{\rm top} } ( S) $ is defined to be the supremum of $ h ( S, \mathfrak A ) $ over all possible $ \mathfrak A $. There is an equivalent definition in the metric case: For a metric $ \rho $, let $ K _ \epsilon ( X, \rho ) $ denote the largest number of points of $ X $ with pairwise distance greater than $ \epsilon $. Then

$$ h _ { \mathop{\rm top} } ( S) = \ \lim\limits _ {\epsilon \rightarrow 0 } \ \lim\limits _ {n \rightarrow \infty } \ { \frac{1}{n} } \mathop{\rm log} \ K _ \epsilon ( X, \rho _ {n} ), $$

where $ \rho _ {n} ( x, y) = \max _ {0 \leq i \leq n } \rho ( S ^ {i} x, S ^ {i} y) $( cf. [2]–[4]).

It turns out that

$$ h _ { \mathop{\rm top} } ( S ^ {n} ) = \ n h _ { \mathop{\rm top} } ( S), $$

and if $ S $ is a homeomorphism, then $ h _ { \mathop{\rm top} } ( S ^ {-1} ) = h _ { \mathop{\rm top} } ( S) $. Hence it is natural to take the topological entropy of a cascade $ \{ S ^ {n} \} $ to be $ h _ { \mathop{\rm top} } ( S) $. For a topological flow (continuous-time dynamical system) $ \{ S _ {t} \} $ it turns out that

$$ h _ { \mathop{\rm top} } ( S _ {t} ) = \ | t | h _ { \mathop{\rm top} } ( S _ {1} ), $$

so it is natural to take the topological entropy of the flow to be $ h _ { \mathop{\rm top} } ( S _ {1} ) $. In a somewhat different way one can define the topological entropy for other transformation groups (it no longer reduces to the topological entropy of one of the elements of the group; cf. [7]).

The topological entropy $ h _ { \mathop{\rm top} } ( S) $ coincides with the supremum of the metric entropy $ h _ \mu ( S) $ over all possible normalized invariant Borel measures $ \mu $( cf. [2], [5]–[7]; [18])

for the existence of $ \max h _ \mu $ and the dependence of $ h _ { \mathop{\rm top} } ( S) $ on $ S $). This is a special case of the variational principle, which establishes a topological interpretation of the value

$$ \sup _ \mu \left [ h _ \mu ( S) + \int\limits f d \mu \right ] $$

for a fixed continuous function $ f $( cf. [4], [8], [9]). The topological entropy gives a characteristic of the "complexity" or "diversity" of motions in a dynamical system (cf. [10], [3], [4]). It is also connected in certain cases with the asymptotics (as $ T \rightarrow \infty $) of the number of periodic trajectories (of period $ \leq T $; cf. Periodic trajectory and [3], [4], [11]–[13]). The "entropy conjecture" asserts that the topological entropy of a diffeomorphism $ S $ of a closed manifold $ W $ is not less than the logarithm of the spectral radius of the linear transformation induced by $ S $ on the homology spaces $ H _ {*} ( W; \mathbf R ) $( cf. [14],[15]). It has been proved in the $ C ^ \infty $- case, .

References

[1]	R.L. Adler, A.G. Konheim, M.H. McAndrew, "Topological entropy" Trans. Amer. Math. Soc. , 114 (1965) pp. 309–319
[2]	E.I. Dinaburg, "On the relation between various entropy characteristics of dynamical systems" Math. USSR Izv. , 5 : 2 (1971) pp. 337–378 Izv. Akad. Nauk SSSR, Ser. Mat. , 35 : 2 (1971) pp. 324–366
[3]	V.M. Alekseev, "Symbolic dynamics" , Eleventh Mathematical Summer School , Kiev (1976) pp. 5–210 (In Russian)
[4]	R. Boywen, "Methods of symbolic dynamics" , Moscow (1979) (In Russian; translated from English) (Collection of papers)
[5]	T.N.T. Goodman, "Relating topological entropy and measure entropy" Bull. London Math. Soc. , 3 (1971) pp. 176–180
[6]	L.W. Goodwyn, "Comparing topological entropy with measure-theoretic entropy" Amer. J. Math. , 94 (1972) pp. 366–368
[7a]	A.T. Tagi-zade, "The entropy of motions of amenable groups" Dokl. Akad. Nauk AzerbSSR , 34 : 6 (1978) pp. 18–22 (In Russian)
[7b]	A.T. Tagi-zade, "Entropy characteristics of amenable groups" Dokl. Akad. Nauk AzerbSSR , 34 : 8 (1978) pp. 11–14 (In Russian)
[8]	A.M. Stepin, A.T. Tagi-zade, "Variational characterization of the topological pressure of amenable transformation groups" Soviet Math. Dokl. , 22 : 2 (1980) pp. 405–409 Dokl. Akad. Nauk SSSR , 254 : 3 (1980) pp. 545–548
[9]	J. Moulin Ollagnier, D. Pinchon, "The variational principle" Studia Math. , 72 (1982) pp. 151–159
[10]	A.A. Brudno, "Entropy and the complexity of the trajectories of a dynamical system" Trans. Moscow Math. Soc. , 44 (1982) pp. 127–152 Trudy Moskov. Mat. Obshch. , 44 (1982) pp. 124–149
[11a]	A.G. Kushnirenko, "Problems in the general theory of dynamical systems on a manifold" Transl. Amer. Math. Soc. , 116 (1981) pp. 1–42 Ninth Math. Summer School (1976) pp. 52–124
[11b]	A.B. Katok, "Dynamical systems with hyperbolic structure" Transl. Amer. Math. Soc. , 116 (1981) pp. 43–96 Ninth Math. Summer School (1976) pp. 125–211
[11c]	V.M. Alekseev, "Quasirandom oscillations and qualitative questions in celestial mechanics" Transl. Amer. Math. Soc. , 116 (1981) pp. 97–169 Ninth Math. Summer School (1976) pp. 212–341
[12]	A.B. Katok, Ya.G. Sinai, A.M. Stepin, "The theory of dynamical systems and general transformation groups with invariant measure" J. Soviet Math. , 7 : 6 (1977) pp. 974–1065 Itogi Nauk. i Tekhn. Mat. Anal. , 13 (1975) pp. 129–262
[13]	A.B. Katok, "Lyapunov exponents, entropy and periodic orbits for diffeomorphisms" Publ. Math. IHES , 51 (1980) pp. 137–173
[14a]	A.B. Katok, "The entropy conjecture" D.V. Anosov (ed.) , Smooth dynamical systems , Moscow (1977) pp. 181–203 (In Russian)
[14b]	M. Shub, "Dynamical systems, filtrations and entropy" Bull. Amer. Math. Soc. , 80 (1974) pp. 27–41
[15]	D. Fried, M. Shub, "Entropy, linearity and chain-recurrence" Publ. Math. IHES , 50 (1979) pp. 203–214
[16]	C. Grilleneberger, "Ergodic theory on compact spaces" , Springer (1976)

Comments

Instead of the collection of translations [4] one may consult [a2].

In the above, $ h _ \mu ( S) $ denotes the entropy of $ S $ with respect to the normalized invariant Borel measure $ \mu $( cf. Entropy theory of a dynamical system). The value $ P _ {S} ( f ) = \sup _ \mu [ h _ \mu ( S) + \int f d \mu ] $, where $ \mu $ runs over the set of all normalized invariant Borel measures, is called the pressure of $ f $( with respect to $ S $). If $ \mu $ satisfies $ h _ \mu ( S) + \int f d \mu = P _ {S} ( f ) $( i.e., the sup is a max), then $ \mu $ is called an equilibrium state or Gibbs measure for $ f $( with respect to $ S $). See [a2], also for existence and uniqueness results.

For recent results about the estimation of topological entropy, see [a1] and the references given there.

References

[a1]	S.E. Newhouse, "Entropy and volume" Ergod. Th. & Dynam. Syst. , 8 (1988) pp. 283–299 Zbl 0638.58016
[a2]	R. Bowen, "Equilibrium states and the ergodic theory of Anosov diffeomorphisms" , Lect. notes in math. , 470 , Springer (1975) Zbl 0308.28010; 2nd ed. (2008) ISBN 978-3-540-77605-5 Zbl 1172.37001