Timeline for Lebesgue entropy zero and positive topological entropy
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 8, 2014 at 20:11 | comment | added | Asaf | "nice enough" - meaning $C^{1+alpha}$ say, with proper algebraic description, the usual examples are hyperbolic toral automorphisms, geodesic flow over homogeneous spaces, and others as well. I think the most recent work in this subject is by Omri Sarig - wisdom.weizmann.ac.il/~sarigo/MP22.pdf . $0$ dimensional support meaning the Hausdorff dimension of the support is $0$. $1$-parameter dynamics meaning that you have here a $\mathbb{N}$ action and not $\mathbb{N}^d$ action for $d>1$. | |
| Feb 8, 2014 at 7:12 | comment | added | shurtados | Thanks for the comment, do you know about references that might help me to understand what you mean by "nice enough"? and also what do you mean by "1 parameter" dynamics and "0 dimensional support" | |
| Feb 7, 2014 at 20:34 | answer | added | rpotrie | timeline score: 3 | |
| Feb 7, 2014 at 15:41 | history | edited | shurtados | CC BY-SA 3.0 |
added 37 characters in body
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| Feb 7, 2014 at 15:40 | comment | added | shurtados | Yes, I'm assuming $f$ is preserving volume, I'll fix that. Thanks. | |
| Feb 7, 2014 at 11:59 | comment | added | Asaf | Regarding the second question, assuming $f$ is "nice enough" (the magic words here are Markov partitions), by the Adler-Weiss theorem, the dynamical system will be isomorphic to a Bernoulli system, hence $\mu$ can be thought of as a measure on a Bernoulli system, and there are plenty of such measures, and in this case the entropy of the measure is related to the dimension of its support, in particular, by using only $1$ parameter dynamics, you cannot rule out $0$-dimensional support. | |
| Feb 7, 2014 at 11:55 | comment | added | Asaf | When you refering to the Lebesgue measure entropy, are you implying tht the Lebesgue measure of the surface is $f$-invariant? | |
| Feb 7, 2014 at 8:35 | history | edited | shurtados | CC BY-SA 3.0 |
edited title
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| Feb 7, 2014 at 8:30 | history | asked | shurtados | CC BY-SA 3.0 |