Transactions of the American Mathematical Society. For the measure of correlations in systems with topological order see Topological entanglement entropy.Bulletin of the London Mathematical Society. "Relating Topological Entropy and Measure Entropy". "RELATIONSHIP BETWEEN TOPOLOGICAL ENTROPY AND METRIC ENTROPY". "Periodic Points and Measures for Axiom A Diffeomorphisms". "Entropy for Group Endomorphisms and Homogeneous Spaces". The use of arbitrary covers yields the same value of entropy. ^ Since X is compact, H( C) is always finite, even for an infinite cover C.For any finite open cover C of X, let H( C) be the logarithm (usually to base 2) of the smallest number of elements of C that cover X. Let X be a compact Hausdorff topological space. Its topological entropy is a nonnegative extended real number that can be defined in various ways, which are known to be equivalent.ĭefinition of Adler, Konheim, and McAndrew An important variational principle relates the notions of topological and measure-theoretic entropy.Ī topological dynamical system consists of a Hausdorff topological space X (usually assumed to be compact) and a continuous self-map f. 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. Later, Dinaburg and Rufus Bowen gave a different, weaker definition reminiscent of the Hausdorff dimension. Their definition was modelled after the definition of the Kolmogorov–Sinai, or metric entropy. For ergodic systems of positive entropy we have the Shannon-McMillan-Breiman theorem which is called the equipartition property 17, 21. Roughly speaking, the number of n-blocks in the subshift is in the order of exp(n). Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew. present a constructive method to build a strictly ergodic subshift of topological entropy dimension for any (0,1). In mathematics, the topological entropy of a topological dynamical system is a nonnegative extended real number that is a measure of the complexity of the system. Loch Ness monsters also appear as translation surfaces with infinite angle or wild singularities, such as the Chamanara or the baker's surface 7, 10, 32, or the infinite staircase 10. For other uses, see Entropy (disambiguation). Topological surfaces of this type are called Loch Ness monsters, and they have appeared in the literature as leaves in foliations by surfaces 19, 30. This article is about entropy in geometry and topology.
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