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We 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)
I 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.
Since 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)
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We 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)
I 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.
Since 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)