# 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)

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.