Timeline for Ergodicity of non-homogeneous "rotations"
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 16, 2012 at 19:19 | history | edited | Ori Gurel-Gurevich | CC BY-SA 3.0 |
deleted 5 characters in body
|
| Feb 16, 2012 at 18:54 | comment | added | Federico | Vaughn, thanks. The wikipedia reference is very much in the direction I was looking for. I also edited as you observed. | |
| Feb 16, 2012 at 18:45 | history | edited | Federico | CC BY-SA 3.0 |
added 2 characters in body
|
| Feb 16, 2012 at 1:07 | comment | added | Vaughn Climenhaga | Also en.wikipedia.org/wiki/Circle_map gives some idea of the richness of behaviour that can happen. | |
| Feb 16, 2012 at 1:07 | comment | added | Vaughn Climenhaga | If I understand your definition, then this is an incredibly large class of systems, and there's essentially no hope of a complete answer. Two remarks jump out: (1) to talk about ergodicity you need an invariant measure. A more meaningful question might be whether or not $f$ is uniquely ergodic. (2) the maps you define are degree 1 maps of the circle. If you put a few more restrictions on $z$ so that they must be homeomorphisms, then there's a very rich theory of orientation preserving homeomorphisms of the circle. See en.wikipedia.org/wiki/Rotation_number, for instance. | |
| Feb 16, 2012 at 1:02 | comment | added | Vaughn Climenhaga | Also your original rotation should be $f(z) = e^{i\theta} z$, and I assume that in your definition of a new map $f$ you want $f(z) = e^{i\theta(z)} z$. | |
| Feb 16, 2012 at 1:01 | comment | added | Vaughn Climenhaga | I guess you mean that $\theta$ can depend on $z$? (Rather than on $\theta$.) | |
| Feb 16, 2012 at 0:52 | history | asked | Federico | CC BY-SA 3.0 |