topological entropy

However, in practice, the Bowen-Dinaburg topological entropy is usually much easier to calculate.

Recently, it was shown that topological orders can also be characterized by topological entropy.

Manning, A.: Topological entropy for geodesic flows.

Rufus Bowen extended this definition of topological entropy in a way which permits X to be noncompact.

Smale horseshoe map is an axiom A diffeomorphism with infinitely many periodic points and positive topological entropy.

Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew.

If the space X is compact and endowed with a topology, or is a metric space, then the topological entropy may also be defined.

If the manifold is nonpositively curved then its volume entropy coincides with the topological entropy of the geodesic flow.

In mathematics, the topological entropy of a topological dynamical system is a nonnegative real number that measures the complexity of the system.

The short form topological entropy is often used, although the same name in ergodic theory refers to an unrelated mathematical concept (see topological entropy).

More generally, volume entropy equals topological entropy under a weaker assumption that M is a closed Riemannian manifold without conjugate points (Freire and Mañé).

Bowen: "Topological Entropy and Axiom A" in Global Analysis (Proceedings of Symposia in Pure Mathematics, vol.

Then the topological entropy of f, denoted h(f), is defined to be the supremum of H(C, f) over all possible finite covers C of X.

The Markov partition thus allows standard techniques from symbolic dynamics to be applied, including the computation of expectation values, correlations, topological entropy, topological zeta functions, Fredholm determinants and the like.

Topological dynamics has intimate connections with ergodic theory of dynamical systems, and many fundamental concepts of the latter have topological analogues (cf Kolmogorov-Sinai entropy and topological entropy).

Given a topologically ordered state, the topological entropy can be extracted from the asymptotic behavior of the Von Neumann entropy measuring the quantum entanglement between a spatial block and the rest of the system.

Bowen's work dealt primarily with axiom A systems, but the methods he used while exploring topological entropy, symbolic dynamics, ergodic theory, Markov partitions, and invariant measures "have application far beyond the axiom A systems for which they were invented."

The second definition clarified the meaning of the topological entropy: for a system given by an iterated function, the topological entropy represents the exponential growth rate of the number of distinguishable orbits of the iterates.

A topological dynamical system consists of a Hausdorff topological space X (usually assumed to be compact) and a continuous self-map f. Its topological entropy is a nonnegative real number that can be defined in various ways, which are known to be equivalent.

The braiding of (2+1) dimensional space-time trajectories formed by motion of physical rods, periodic orbits or "ghost rods", and almost-invariant sets has been used to estimate the topological entropy of several engineered and naturally occurring fluid systems, via the use of Nielsen-Thurston classification.