"Minimal extensions of smooth dynamical systems"Dirbák, MatúšA classical result of Fathi and Herman states that a smooth compact connected manifold $Z$ without boundary admitting a locally free action of $\mathbb T^1$ (respectively, an almost free action of $\mathbb T^2$) admits a minimal diffeomorphism (respectively, a minimal flow $\mathbb R\curvearrowright Z$). Proofs of these results rely on a variation of the approximation by conjugation method which is due to Anosov and Katok. In the first part of our talk we will give examples of manifolds $Z$ satisfying the assumptions formulated above in the class of homogeneous spaces of compact connected Lie groups. In the second part we will be interested in the existence of minimal skew products over minimal flows having a free cycle. We say that a minimal flow $\mathcal F\colon\Gamma\curvearrowright X$, whose acting group $\Gamma$ is a connected Lie group and the phase space $X$ is a compact manifold, has a free cycle if $\text{rank}(\text{im}(f))<\text{rank}(H_1^w(X))$, where $f\colon H_1^w(\Gamma)\to H_1^w(X)$ is the morphism induced by a transition map of $\mathcal F$ and $H_1^w(\Gamma)$, $H_1^w(X)$ are the first weak homology groups of $\Gamma$, $X$, respectively (these are obtained from the ordinary homology groups by factoring out their torsion subgroups). We show that if $X$ and $\mathcal F$ are smooth/analytic, $\mathcal F$ has a free cycle and $\ell\in\mathbb N$ then $\mathcal F$ has an abundance (in the algebraic sense) of minimal group extensions with the fibre $\mathbb T^{\ell}$ which are also smooth/analytic. We use this result together with ideas of Fathi and Herman to show that such flows admit minimal smooth skew products with the fibre $Z$ being a compact manifold admitting an almost free action of $\mathbb T^2$. In the last part of our talk we will give examples of minimal flows with free cycles, mostly in the class of homogeneous flows of connected Lie groups (which are analytic). |
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