Timeline for Characterising ergodicity of continuous maps
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Feb 24, 2013 at 16:52 | comment | added | Ian Morris | @Julian: Every invariant probability measure of a minimal transformation is fully supported, because otherwise its support would be a nonempty closed invariant proper subset, contradicting minimality. So the two statements are equivalent. | |
| Feb 24, 2013 at 14:38 | comment | added | Julian Newman | @Ian and Anthony: Just to be clear, I did not say that I require every invariant probability measure of a minimal transformation to be ergodic - I just required that every strictly positive invariant probability measure of a minimal transformation had to be ergodic. (By strictly positive, I mean that its support is the whole of $X$). Is this still equivalent to requiring that every minimal transformation is uniquely ergodic? (And in the case $X=\mathbb{R}^n$, if the requirement still is not satisfied, what about if we weaken the requirement by restricting to, say, diffeomorphisms on $X$?) | |
| Feb 24, 2013 at 12:08 | comment | added | Ian Morris | @Julian: this is equivalent to asking for a condition on $X$ such that every minimal transformation on $X$ is uniquely ergodic, i.e. has only one invariant measure. (If a transformation has two distinct invariant measures then a strict linear combination of the two is never ergodic.) Such conditions do exist: finite spaces $X$ have this property, as does the circle (I think) but as Anthony says this is a severly restrictive requirement. The broader stroke of your question seems to be whether ergodicity can be easily characterised using only topological concepts. The answer to this is "No". | |
| Feb 24, 2013 at 3:26 | comment | added | Anthony Quas | @Julian: This is hopeless. On any reasonable space, there will be transformations that are minimal, but not strictly ergodic. | |
| Feb 24, 2013 at 2:44 | comment | added | Julian Newman | Thank you. This is most helpful. Do you know any conditions on $X$ under which, if $\mu$ is a strictly positive probability measure on $X$, then every minimal $\mu$-preserving continuous transformation is ergodic? (E.g. is this true for Euclidean space $X=\mathbb{R}^n$?) | |
| Feb 24, 2013 at 2:25 | vote | accept | Julian Newman | ||
| Feb 24, 2013 at 1:35 | history | answered | Ian Morris | CC BY-SA 3.0 |