Talk:Topological entropy
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Measure, metric and topology
editWe have entropy defined using a measure. We have entropy defined using topology. And we have entropy defined using a metric. Why do we keep calling the "measure-theoretic" entropy as "metric-entropy"? I suggest changing it to "Kolmogorov-Sinai entropy", or even "measure-theoretic entropy". — Preceding unsigned comment added by André Caldas (talk • contribs) 00:47, 22 October 2012 (UTC)
Low and high entropy
editI just copied this article from Planet math, so am not to clear on its interpretation. Very curiously, it seems to be saying that ergodic systems have a very low entropy (!), while only dissipative systems would have a high entropy. Curious. linas 14:19, 7 June 2006 (UTC)
- Never mind, I misread one of the lines. None-the-less, some examples would be good. linas 14:31, 7 June 2006 (UTC)
- The "metric" is not well-defined because is not supposed to be injective.
Kolmogorov-Sinai entropy
editSince I last looked at this article, a section was added called Definition of Adler, Konheim, and McAndrew but the definition given there seems to be identical, at least to my tired eyes, to the definition of Kolmogorov-Sinai entropy. Now the lead explains that this is somehow an improvement, but I don't see quite what the difference is ... Soo .. what's up with that? linas (talk) 03:56, 22 November 2010 (UTC)
- Kolmogorov–Sinai entropy is measure-theoretic, i.e. it depends on the invariant measure μ, whereas the topological entropy is purely topological, i.e. it depends only on the topological conjugacy class of the map T. There is a very important relation between the two notions, the variational principle: htop(T) is the supremum over all invariant measures of hμ(T). You can read the details in the Scholarpedia article. Arcfrk (talk) 14:25, 22 November 2010 (UTC)