# Topological entropy

A concept in topological dynamics and ergodic theory analogous to the metric entropy of dynamical systems (introduced in ). 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. ).

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. ).

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. , ; )

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. , , ). The topological entropy gives a characteristic of the "complexity" or "diversity" of motions in a dynamical system (cf. , , ). 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 , , ). 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. ,). It has been proved in the $C ^ \infty$- case, .

How to Cite This Entry:
Topological entropy. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topological_entropy&oldid=50928
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article