Abstract: This is a joint result obtained with J. Janczewska and M. Izydorek from Gdańsk University of Technology and E. Stepanov from University of Pisa.
First of all, we mention the fact that having a conservative flow on Euclidian space with the so-called zero mean drift condition, we can make all the points of the phase space non-wandering simultaneously by adding an arbitrarily C^1 small perturbation. This will give, in particular, chain transitivity of the flow and validity of the Pugh's closing lemma.
Secondly, we briefly discuss the opportunity to prove an analog of Connecting Lemma.
Finally, we consider the set of vector fields such that the Lie algebra engendered by these vector fields spans the whole space (the so-called Hörmander condition). We suppose that this condition is uniform and the considered system of o.d.es satisfies assumptions of the first part. Then, any points of the phase space can be linked by a horizontal pseudotrajectory (the corresponding control is of the linear hull of the considered vector fields).
Roughly speaking, we discuss an opportunity to drive a car on an icy lake with a strong wind.
This is a 'dynamical version' of the famous Chow-Rashevskii's theorem. The 'horizontal' versions of C^0 closing and connecting lemmas will also be discussed.
