Minimal elements of minimal R^k actions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T13:39:28Z http://mathoverflow.net/feeds/question/29061 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29061/minimal-elements-of-minimal-rk-actions Minimal elements of minimal R^k actions coudy 2010-06-22T08:39:02Z 2010-07-07T19:38:45Z <p>C. Pugh and M. Shub showed in 1971 that, given an ergodic action of $G=\mathbb{R}^k$ on some separable finite measure space $(X,\mu)$, then all elements of $G$ , off a countable family of hyperplanes, are ergodic.</p> <p>Is there an analogous statement in the topological setting, with ergodicity replaced by minimality (i.e. all orbits are dense) and $X$ assumed to be compact ?</p> http://mathoverflow.net/questions/29061/minimal-elements-of-minimal-rk-actions/30945#30945 Answer by coudy for Minimal elements of minimal R^k actions coudy 2010-07-07T19:38:45Z 2010-07-07T19:38:45Z <p>A colleague pointed out the following counterexample. Let $h_t$ be the horocyclic flow on a negatively curved compact surface S. This R action is known to be minimal. Now Consider the $R^2$ action on $S\times S$ given by $(s,t)\rightarrow (h_s,h_t)$. This action is again minimal. </p> <p>The action of the diagonal $\{(s,s), s\in R\}$ is not minimal since the orbit of any point (x,x) stays in the diagonal. </p> <p>Let $\theta\in R$. The action of the line $\{(s,\theta s), s\in R\}$ is not minimal because it is conjugated to the diagonal action. This comes from the fact that the two actions $h_{\theta s}$ and $h_s$ are conjugated by the geodesic flow. </p> <p>As a result, there are no elements in $R^2$ acting minimally, although $R^2$ itself acts minimally.</p>