A concept in topological dynamics and ergodic theory analogous to the metric entropy of dynamical systems (introduced in ). For an open covering of a compactum , let denote the logarithm (usually to base 2) of the smallest number of elements of the covering that can cover . If is a continuous mapping, then the limit
exists, where is the covering whose elements are all the non-empty intersections of the elements of and . The topological entropy is defined to be the supremum of over all possible . There is an equivalent definition in the metric case: For a metric , let denote the largest number of points of with pairwise distance greater than . Then
It turns out that
so it is natural to take the topological entropy of the flow to be . 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. ).
for the existence of and the dependence of on ). This is a special case of the variational principle, which establishes a topological interpretation of the value
for a fixed continuous function (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 ) of the number of periodic trajectories (of period ; cf. Periodic trajectory and , , –). The "entropy conjecture54C70entropy conjecture" asserts that the topological entropy of a diffeomorphism of a closed manifold is not less than the logarithm of the spectral radius of the linear transformation induced by on the homology spaces (cf. , ). It has been proved in the -case, .
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In the above, denotes the entropy of with respect to the normalized invariant Borel measure (cf. Entropy theory of a dynamical system). The value , where runs over the set of all normalized invariant Borel measures, is called the pressure of (with respect to ). If satisfies (i.e., the sup is a max), then is called an equilibrium state or Gibbs measure for (with respect to ). See [a2], also for existence and uniqueness results.
For recent results about the estimation of topological entropy, see [a1] and the references given there.
|[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|
Topological entropy. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Topological_entropy&oldid=42692