Timeline for When is there a natural Riemannian metric whose measure preserves a self-diffeomorphism?
Current License: CC BY-SA 2.5
16 events
when toggle format | what | by | license | comment | |
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Oct 13, 2010 at 15:18 | answer | added | Stephen Miller | timeline score: 1 | |
Sep 9, 2010 at 1:39 | answer | added | Martin | timeline score: 2 | |
Jun 10, 2010 at 2:57 | comment | added | Steve Huntsman | Wish I could accept both answers, and Deane's comments as well. Thanks all! | |
Jun 10, 2010 at 2:56 | vote | accept | Steve Huntsman | ||
Jun 9, 2010 at 17:04 | comment | added | Steve Huntsman | ...continuing...but now it seems that I have any right to expect the situation to be anywhere near my hopes. | |
Jun 9, 2010 at 17:02 | comment | added | Steve Huntsman | @Deane--Very good point, I (wasn't aware of and) hadn't considered the uniqueness of the diffeomorphism, esp. vs. the conformal rescaling....In the application I have in mind, $f$ is Anosov and the only natural initial choice of metric I can think of would be an Anosov (by normalization, not necessarily Lyapunov) metric. But this is not (unique as far as I know or) $f$-invariant, even assuming $f$ is conservative. But what I mean by "natural" can be somewhat clarified in the ideal case: there'd be some unique way to pick out a distinguished metric giving rise to the $f$-invariant measure. | |
Jun 9, 2010 at 16:39 | comment | added | Deane Yang | Why is pulling back $g$ using a diffeomorphism constructed using Moser's trick more natural or better than just changing $g$ by a conformal factor? Given the original metric $g$, the conformal factor is uniquely determined (which fits my definition of "natural"), but the diffeomorphism that pulls back $\nu$ to $\nu_f$ is far from unique. | |
Jun 9, 2010 at 13:10 | comment | added | Steve Huntsman | I thought the pullback suggested by the Moser theorem in coudy's answer was pretty close to what I had in mind. | |
Jun 9, 2010 at 12:00 | comment | added | Deane Yang | Steve, as I comment below, there are many different metrics with a given measure. All you have to do is start with arbitrary metric $g$ with volume measure $\mu$ and define $g_f = g(\nu/\mu)^{2/n}$. Again, what do you mean by "natural"? | |
Jun 8, 2010 at 21:12 | comment | added | Steve Huntsman | So-called conservative diffeomorphisms preserve a natural Riemannian measure. So my question could be rephrased as: given a conservative $f$ and its preserved measure $\nu$, is there a natural metric $g_f$ whose measure is also $\nu$? | |
Jun 8, 2010 at 21:04 | answer | added | coudy | timeline score: 6 | |
Jun 8, 2010 at 20:48 | answer | added | rpotrie | timeline score: 3 | |
Jun 8, 2010 at 20:38 | comment | added | Steve Huntsman | @Will--For background: arxiv.org/pdf/0804.0167 | |
Jun 8, 2010 at 20:36 | comment | added | Steve Huntsman | @Deane--The original metric is just mentioned to distinguish it from the putative $g_f$. | |
Jun 8, 2010 at 19:56 | comment | added | Deane Yang | What is the role of the original metric $g$ in your question? And what is the meaning of "natural"? | |
Jun 8, 2010 at 19:52 | history | asked | Steve Huntsman | CC BY-SA 2.5 |